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WORKS  OF 
PROFESSOR  C  W.  MACCORD 

PUBLISHED   BY 

JOHN  WILEY  &   SONS.  INC. 


Doscriptive  Geometry. 

With  Applications  to  Isometrical  Drawing  and  Cavalier 
Projection.  8vo.  vi  -|-  248  pages,  280  figures.  Cloth, 
$3.00. 

Kinematics;  or,  Practical  Mechanism. 

8vo.     xi-f335  pages,  306  figures.     Cloth,  $5.00, 

Mechanical  Drawing. 

Part    I.     Progressive  Exercises. 

Part  II.     Practical  Hints  for  Draughtsmen. 

The   two   parts   complete    in   one  volume.      4to.      258 

pages,  232  figures.     Cloth,  $4.00. 

Velocity  Diagrams. 

Their  Construction  and  their  Uses.  Addressed  to  all 
those  interested  in  Mechanical  Movements.  8vo.  iii-|- 
116  pp.,  83  figures.     Cloth,  $1.50. 


ELEMENTS 


OP 


DESCRIPTIVE  GEOMETRY. 


WITH  APPLICATIONS  TO 


ISOMETRICAL  DRAWING  AUD  CAVALIER  PROJECTION. 


BY 


CHARLES   WILLIAM   MacCORD,  A.M.,  Sc.D., 

Emeritus  Professor  of  Mechanical  Drawing  and  Designing  in  the  Stevens  Institute  of 

Technoloffy,  Hohoken,  N.  J. ; 

Member  of  the  American  Society  of  Mechanical  Engineers; 

Formerly  Chief  Draughtsman  for  Capt.  John  Ericsson  ; 

Author  of  "  Kinematics,''  "  Mechanical  Drawing  :   Progressive  Exercises  and  Practictd 

Hints,''  "  Velocity  Diagrams,"  and  Numerous  Monographs  on  Mechanism. 


SECOND  EDITION,  REVISED. 
EIGHTH   THOUSAI^D. 


NEW  YORK: 
JOHN  WILEY  &  SONS,  Inc. 
London:   CHAPMAN  &  HALL,  Limited 
1914 


M  3 


Copyright,  1895, 

BY 

CHARLES  WILLIAM  MacCORD. 


THE  SCIENTIFIC   PRESS 

ROBERT   DRUMMONO   ANO    COMPANr 

BROOKLYN,    N.   Y. 


PREFACE. 


Having  been  convinced,  by  a  class-room  experience  of  many 
years  at  the  Stevens  Institute  of  .Technology,  of  the  desirability  of  a 
text-book  on  Descriptive  Geometry  different  in  some  respects  from 
any  previously  existing,  I  have  endeavored  to  produce  a  work  suit- 
able for  use  in  colleges  and  scientific  schools,  and  also  by  those  who 
may  wish  to  acquire  some  knowledge  of  the  subject  without  the  aid 
of  an  instructor.  In  the  course  of  that  experience  many  points 
have  arisen,  leading  to  original  work  embodied  in  this  treatise;  in 
the  preparation  of  which,  however,  much  benefit  has  been  derived 
from  reference  to  the  works  of  Olivier,  Jullien,  Church,  Warren, 
Watson,  and  others. 

The  study  of  Descriptive  Geometry  is  not  usually  begun,  nor 
should  it  ever  be,  until  some  familiarity  with  the  ordinary  opera- 
tions of  Mechanical  Drawing  has  been  attained.  But  when  the 
former  is  taken  tip  its  identity  with  the  latter  should  never  be  lost 
sight  of,  as  it  too  often  is :  for  this  reason  a  departure  has  been 
made  from  the  stereotyped  methods  of  treatment,  which  in  fact 
tend  rather  to  conceal  than  to  exhibit  tliat  identity. 

At  the  outset  considerable  difficulty  is  often  experienced  in 
forming  clear  conceptions  of  the  relations  between  abstract  things, 
such  as  lines  and  planes,  by  the  aid  of  orthographic  projections 
only.  The  power  of  doing  so  is  of  course  essential;  and  it  is 
believed  that  the  pictorial  representations  which  have  been  intro- 
duced will  be  of  assistance  in  acquiring  it.  But  that  power  will  be 
best  developed,  and  greatly  increased,  by  the  instrumental  construc- 
tion of  the  problems — which  indeed  is  absolutely  necessary  to  the 

iii 

359450 


IV  PREFACE. 


attainment  of  such  a  mastery  of  the  principles  and  processes  as 
alone  would  be  of  any  practical  value. 

As  a  hint  to  those  who  may  choose  to  dispense  with  an  instruc- 
tor, it  may  be  stated  that  at  the  Stevens  Institute  of  Technology  it 
is  required  that  tlie  diagrams  shall  be  drawn  with  care,  but  not 
required  that  they  shall  be  drawn  in  ink.  JS^or  is  the  larter  recom- 
mended; the  time  required  to  ink  in  one  diagram  can  be  better 
occupied  in  drawing  another;  moreover,  work  of  this  description 
affords  the  best  of  practice  in  neat,  effective,  and  accurate  pen- 
cilling,— an  accomplishment  which  is  becoming  more  and  more 
important  to  the  practical  draughtsman. 

C.  W.  JVIacCord, 

HOBOKEN,  New  Jersey,  September  23,  1895. 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

PAGB 

Definitions. — The  Principal  Planes  of  Projection. — The  Four  Dihedral 
Angles. — The  Profile  Plane. — Representation  of  the  Point,  Right 
Line,  and  Plane.— Geometrical  Principles  and  Deductions. — Revolu- 
tion and  Counter-revolution.  —  Separate  Construction  of  the  Hori- 
zontal and  Vertical  Projections.  —  Supplementary  Planes  and 
Projections 1 

CHAPTER  II. 
Elementary  Problems  Relating  to  the  Point,  Right  Line,  and  Plane     .     .    32 

CHAPTER  III. 

Generation  and  Classification  of  Lines  and  Surfaces.  —  Tangents, 
Normals,  and  Asymptotes  to  Lines.  —  Osculation,  Rectification, 
Radius  of  Curvature. — Tangent,  Normal,  and  Asymptotic  Planes 
and  Surfaces 69 

CHAPTER  IV. 

On  the  Determination  of  Planes  Tangent  to  Surfaces  of  Single  and  of 
Double  Curvature 93 

CHAPTER  V. 

Of  Intersections  and  Developments 113 

Intersection  of  Surfaces  by  Planes.— Development  of  Single-curved 
Surfaces. — Tangents  to  Curves  of  Intersection  before  and  after 
Development.  —  Problem  of  the  Shortest  Path.  —  Intersections  of 
Single-curved  Surfaces. — Infinite  Branches. — Intersections  of  Double- 
curved  Surfaces. — Intersection  of  a  Cone  with  a  Sphere.—  Develop- 
ment of  the  Oblique  Cone. 

V 


VI  TABLE   OF   CONTENTS. 

CHAPTER  VI. 

PAGE 

Of  Warped  Surfaces 152 

The  Hyperbolic  Paraboloid  ;  its  Vertex,  Axis,  Principal  Diametric 
Planes,  and  Gorge  Lines.— The  Conoid. — The  Hyperboloid  of  Revolu- 
tion.— The  Elliptical  Hyperboloid,  and  its  Analogy  to  the  Hyperbolic 
Paraboloid. — The  Helicoid  of  Uniform  and  of  Varying  Pitch. — The 
Cylindroid. — The  Cow's  Horn.  —  Warped  Surfaces  of  General 
Forms. — Planes  Tangent  to  Warped  Surfaces. — Warped  Surfaces 
Tangent  to  Each  Other. — Intersections  of  Warped  Surfaces. 

CHAPTER  VII. 
Isometrical  Drawing,  Cavalier  Projection,  and  Pseudo-perspective  .    .    .  230 


DESCRIPTIVE  GEOMETRY. 


CHAPTER  I. 

DEFINITIONS. THE     PRINCIPAL     PLANES     OF     PROJECTION. THE     FOUR 

DIHEDRAL    ANGLES. THE  PROFILE    PLANE. REPRESENTATION  OF 

THE      POINT,     OF      THE     RIGHT      LINE,     AND      OF     THE      PLANE. ■ 

GEOMETRICAL     PRINCIPLES     AND    DEDUCTIONS. REVOLUTION     AND 

COUNTER-REVOLUTION. SEPARATE     CONSTRUCTION    OF    THE    HORI- 
ZONTAL   AND    VERTICAL     PROJECTIONS. SUPPLEMENTARY    PLANES 

AND    PROJECTIONS. 

1.  Descriptive  Geometry  treats  of  tlie  methods  of  making, 
Avitli  nmthematical  exactness,  drawings  for  the  representation  not 
only  of  geometrical  magnitudes,  but  of  the  solutions  of  problems 
relating  to  them. 

2.  This  branch  of  science  does  not  deal  with  the  phenomena  of 
t)inocular  vision,  and  for  its  purposes  the  eye  is  regarded  as  a  single 
point. 

The  surface  upon  which  a  drawing  is  made  may  be  of  any 
form,  as  cylindrical,  in  panoramic  painting,  or  spherical,  in  deco- 
rating the  interior  of  a  dome.  But  in  order  to  make  correct  draw- 
ings upon  such  surfaces,  it  is  necessary  to  be  thoroughly  familiar 
Avith  the  methods  of  making  them  upon  planes,  which  are  usually 
employed ;   and  to  these  our  attention  will  be  confined. 

3.  The  object  to  be  drawn  may  be  placed  between  the  eye  and 
the  plane,  or  the  plane  may  be  placed  between  the  eye  and  the 
object.  In  either  case,  light  is  reflected  from  any  point  of  the 
object  to  the  eye  in  a  right  line ;  and  the  point  in  which  that  line, 
produced  if  necessary,  pierces  the  plane,  is  the  representation  of 


2  b'ESCRTPflVE   GEOMETRY. 

that  point  in  tlie  object  from  wliicli  it  came.  A  sufficient  nnin])ei 
of  such  points  being  found,  the  outhnes  maj  be  fullj  determined ; 
and  the  drawing  thus  made  will  present  to  the  eye,  if  placed  in  the 
position  originally  assigned  to  it,  the  same  appearance  as  the  actual 
contour  of  the  object  itself. 

It  may  be  said,  then,  that  the  representation  of  a  point  is  found 
by  projecting  it  along  a  right  line  passing  through  the  eye.  Sucli 
a  line  is  called  a  projecting  line,  and  all  drawings  thus  made  are 
technically  called  projections. 

4.  If  the  eye  is  at  a  Unite  distance,  the  drawing,  on  any  surface, 
is  called  a  scenographic  projection.  If  made  upon  a  vertical  plane, 
against  winch  the  eye  is  directed  jperpendiciilarly ,  the  drawing  is 
said  to  be  in  perspective;  the  plane  is  then  called  the  picture  pl&ne, 
and  the  projecting  lines,  which  converge,  are  called  visual  rays. 

If  the  eye  is  removed  to  an  infinite  distance,  the  i3rojecting  lines 
become  parallel  to  each  other  and  to  the  axis  of  vision.  The  plane 
upon  which  the  drawing  is  made  is  called  the  plane  of  projection; 
it  may  be  perpendicular  to  tlie  projecting  lines,  in  wliicli  case  the 
drawing  is  called  an  orthographic  projection;  or  it  may  cut  them 
obliquely,  and  the  drawing  is  then  called  an  oblique  projection. 

5.  Of  these  three  systems  of  projection  the  second  is  the  most 
simple  and  the  most  extensively  used,  and  a  knowledge  of  it  is  an 
essential  preliminary  to  the  study  of  the. others.  We  proceed  then 
at  present  to  consider  the  methods  of  representing  magnitudes  and 
the  solution  of  problems  in  orthographic  projection  only.  Evidently 
the  number  of  such  projections,  or  views,  necessary  to  tlie  adequate 
representation  of  an  object  of  three  dimensions,  will  depend  mucli 
on  the  form  of  the  object  itself.  But,  beginning  with  the  least  of 
geometrical  magnitudes,  the  point,  considered  as  a  visible  and 
material  particle ;  it  can  be  located  in  space  by  giving  its  distance 
from  each  of  two  fixed  planes,  and  represented  by  its  projections 
upon  them. 

6.  The  Principal  Planes  of  Projection.  The  most  simple  and 
natural  relation  between  two  planes  for  this  purpose,  which  is 
universally  adopted,  is  that  shown  in  Fig.  1 ;  the  one  being  hori- 
zontal, the  other  vertical.  Moreover,  these  suffice  for  many, 
though  by  no  means  all,  of  the  ordinary  operations  of  descriptive 


DESCRIPTIVE    GEOMETRY. 


geometry ;   lience  we  may  say  that  the  principal  planes  of  projection 


are 


1.  The  horizontal  plane,  usually  designated  simply  as  H  for 
brevity. 

2.  The  vertical  plane,  usually  designated  simply  as  T  for  brevity. 

These  planes  are  supposed  to  extend  indefinitely  in  each  direc- 
tion ;  they  intersect  in  a  line  called  the  ground  line,  designated  and 
referred  to  as  AB.  The  eye  is  supposed  to  be  at  an  infinitely  re- 
mote point,  in  front  of  the  vertical  plane  and  above  the  horizontal 
plane  ;  whence  it  is  directed  either  perpendicularly  against  T,  as 


V 

2 

1 

•p 

1 

H                   30 
V 

4         i 

Fig. 

^       H 

2 

ip 

Tig.  3 


d'    ^-^ 
0      \     f 

i   d 

E* 

c    1 

e' 

Fig.  4 


i.     2. 

3. 

4. 

d\ 

et 

1         1 

i 

A  i 

1 

■et 

B 

c* 

1 

ft 

/I 

64 

/i 

Fig.  5 


-■^E,, 


Fig.  6 


shown  by  the  horizontal  arrow,  or  perpendicularly  downward  upon 
H,  as  sliown  by  the  vertical  arrow. 

7.  The  Four  Angles.  Fig.  1  is  a  pictorial  representation  of  a 
model,  such  as  can  readily  be  made  by  cutting  tAVO  cards,  each 
through  half  its  length,  and  "halving"  them  together.  If  this 
model  be  lield  so  that  the  eye  is  directly  in  front  of  the  point  A, 
and  looking  in  the  direction  AB,  it  will  appear  as  shown  in  Fig.  2 ; 
the  ground  line  appearing  as  the  point  0,  while  each  plane,  being 


4  DESCRIPTIVE   GEOMETRY. 

seen  edgewise,  will  appear  as  a  mere  line.     Thus  the  two  planes 
form  four  equal  dihedral  angles,  which  are  numbered  1,  2,  3,  4,  in 
the  order  shown;   that  in  which  the  eye  is  placed,   as  above  set 
forth,  being  the  Jirst  angle. 
Thus  we  have : 

1st  Angle Above  H  and  in  front  of  V„ 

2d  Angle Above  H  and  behind  V. 

3d  Angle Below  H  and  behhid  V. 

4th  Ano:le Below  H  and  in  front  of  V. 


't5^ 


8.  In  Figs.  1  and  2,  P  is  a  point  in  space,  here  taken  in  the 
first  angle ;  the  vertical  line  Pjp  is  its  horizontal  projecting  line,  and 
jp  is  its  horizontal  projection.  The  line  Pj)'  perpendicular  to  V  is 
its  vertical  projecting  line,  and^'  is  its  vertical  projection. 

This  illustrates  the  notation  adopted,  the  capital  letter  denoting 
a  point  in  space,  and  the  small  letters  denoting  its  projections,  that 
npon  the  vertical  plane  being  accented :  thus  we  write,  for  ex- 
ample, "  the  point  J/,"  indicating  the  point  whose  horizontal  pro- 
jection is  w,  and  whose  vertical  projection  is  m' . 

These  two  projections  suffice  to  detennine  the  position  of  the 
point  in  space ;  for  if  in  Fig.  1  we  suppose  ^j>  and  2^'  only  to  be 
given,  a  vertical  line  through  _^  and  a  perpendicular  to  Y  through 
^'  will  intersect  in  P, 

9.  In  Fig.  1,  draw  through  J9  a  perpendicular  to  V,  cutting  AB 
in  c ;  then  jpc  is  parallel  and  equal  to  Pjp' .  Completing  the  rect- 
angle, ;p'g  is  parallel  and  equal  to  Pp.  That  is  to  say,  the  distance 
of  a  point  in  space  from  the  vertical  plane  is  equal  to  the  distance 
of  its  horizontal  projection  from  the  ground  line ;  and  its  distance 
from  the  horizontal  plane  is  equal  to  tlie  distance  of  its  vertical  pro- 
jection from  the  ground  line. 

10.  Hold  the  page  in  a  vertical  position,  and  looking  perpeiN 
dicularly  against  it  at  Fig.  3,  suppose  the  paper  to  be  the  plane  ] 
of  Fig.  1 ;  the  line  AB  will  then  represent  the  horizontal  plane 
seen  edgewise,  and  the  line  p'c  will  be  seen  in  its  true  length  and 
position.  Next,  hold  the  page  in  a  horizontal  position,  and  look- 
ing vertically  downward  at  the  same  figure,  imagine  the  paper  to 
represent  the  plane  II  oi  Fig.  1.     The  line  AB  will  then  represent 


DESCRIPTIVE   GEOMETRY.  5 

the  vertical  plane  seen  edgewise,  and  the  line  ^c  will  be  seen  in  its 
true  length  and  position.  Thus  the  single  plane  of  the  paper  rep- 
resents both  H  and  V,  and  with  a  little  effort  can  at  pleasure  be  re- 
garded as  either  the  one  or  the  other.  Fig.  3,  then,  is  a  repre- 
sentation of  a  point  P,  situated  in  the  Urst  angle,  in  orthographic 
projection  upon  the  tw^o  principal  planes.  In  reality,  only  the 
ground  line  and  the  two  points^  and^i'  are  absolutely  necessary; 
but  we  observe,  tliat  since  in  Fig.  1  jpc  and  p'c  are  both  perpen- 
dicular to  ^^  at  the  same  point  <?,  they  will  in  Fig.  3  coincide  in 
one  right  line.  That  is  to  say,  the  two  projections  of  a  given  point 
mnst  lie  on  the  same  perpendicular  to  the  ground  line. 

11.  The  Profile  Plane.  In  effect.  Fig.  3  is  both  sl  front  yiew 
and  a  toj?  view  of  the  model  shown  in  Fig.l ;  the  space  above  AB 
representing  the  upper  part  of  Y  and  also  that  part  of  H  which  is 
behind  V,  wliile  the  space  below  AB  represents  the  lower  part  of  V 
and  the  front  part  of  H. 

]^ow.  Fig.  2  is  an  end  view  of  the  same  model,  the  eye  looking 
(7)  in  the  direction  J. ^ ;  and  J.^  is  seen  in  Fig.  l.to  be  perpen- 
dicular to  tlie  plane  of  the  rectangle  Pc :  in  other  words.  Fig.  2  is 
an  orthographic  projection  upon  a  plane  perpendicular  to  the 
ground  line. 

Such  a  plane  is  called  a  profile  plane ;  the  projection  upon  it  is 
called  simply  a  profile,  and  if  made,  as  here,  separate  and  distinct 
from  the  projections  upon  the  prmcipal  planes,  it  is  often  of  the 
greatest  use. 

Evidently,  a  profile  may  be  constructed,  representing  the  model 
as  seen  from  the  right,  looking  in  the  direction  BA ;  the  first  and 
fourth  angle  will  in  that  case  lie  on  the  left  of  V,  and  the  order  of 
the  numbers  will  be  the  reverse  of  that  shown  in  Fig.  2. 

12.  Location  of  the  Profile.  If  the  profile  is  made  on  the  first 
supposition,  (7)  the  view  heingfro77i  the  left,  it  should  be  placed 
at  the  left  of  the  drawing  showing  the  projections  on  the  principal 
planes ;  if  made  on  the  second  supposition,  that  the  view  is  from 
the  right,  the  profile  should  be  placed  at  the  right  of  that  drawing. 
Thus,  Fig.  5  shows  the  projections  on  H  and  V  of  four  points, 
one  in  each  dihedral  angle ;  Fig.  4  is  a  profile  showing  the  same 
points,  with  their  projecting  lines,  seen  from  the  left ;   and  Fig.  6 


b  DESCKirXIVE   GEOMETRY. 

is  a  profile,  in  wliicli  the  eve  is  at  the  right,  looking  in  the  direc- 
tion jSA. 

13.  If  in  Fig.  4  we  suj)pose  the  point  C,  for  instance,  to  ap- 
proach the  vertical  plane,  Co'  will  be  diminished,  and  c  will  ap- 
proach 0 ;  w^hen  the  point  reaches  T ,  Ce  will  coincide  with  c'  0. 
Eeasoning  similarly  vrith  regard  to  H,  we  perceive  that  if  a  poiot 
lies  in  either  plane,  it  will  be  its  own  projection  on  that  plane,  and  its 
projection  on  the  other  plane  will  lie  in  the  ground  line.  If  it  lies  in 
both  planes,  the  point  itself  and  both  its  projections  coincide  in  one 
point  on  the  ground  line. 

In  Fig.  5  it  is  observed  that  both  projections  of  D,  a  point  in 
the  second  angle,  lie  above  AB,  while  both  those  of  J^^,  which  is  in 
the  fourth  angle,  lie  below  AB. 

Now,  were  D  equally  distant  from  H  and  V,  its  j^i-ojections  d 
and  d'  would  fall  together  in  one  ]3oint;  but  the  t  ,o  letters  would 
still  be  used.  The  same  would  in  like  case  be  true  of  the  projec- 
tions of  J^;  and  conversely,  if  the  two  projections  coincide,  but  do 
not  lie  on  the  ground  line,  the  point  itself  is  equidistant  from  H  and 
V,  and  therefore  lies  in  a  jAeme  bisecting  the  second  and  fourth 
angles. 

EEPRESENTATION    OF    THE    EIGHT    LINE. 

14.  The  projection  of  any  line  upon  any  plane  is  determined  l)y 
projecting  all  its  points  upon  that  j^lane.  In  the  case  of  a  right 
line,  two  points  determine  it  in  space,  and  the  projections  of  these 
two  are  sufficient. 

Thus  in  Fig.  7,  m  and  n  are  the  projections  of  Jf  and  i\^upon 
the  plane  JTY.  Tlie  two  projecting  perpendiculars  determine  a 
plane  containing  the  line  J/iV^;  this  is  called  tlie  projecting  plane, 
and  it  cuts  xy  in  a  right  line  wai,  called  its  trace,  which  is  the  pro- 
jection of  the  given  line. 

If  the  line  is  not  parallel  to  the  plane,  it  must  pierce  it  when 
prolonged.  The  point  of  penetration  must  lie  on  tlie  given  line, 
and  also  in  its  projection ;  it  is  therefore  at  their  intersection  P. 
This  point  is  sometimes  called  the  trace  of  the  line  on  the  plane. 

Draw,  in  the  projecting  plane  J[/??,  a  line  iVT  perpendicular  to 
Mm ;  it  is  also  .perpendicular  to  JVn,  consequently  In  is  a,  rectangle. 


DESCRIPTIVE   GEOMETRY.  7 

Now  regarding  IN  as  tlie  given  line,  mn  is  its  projection ;  there- 
fore, if  a  line  be  parallel  to  a  plane,  its  projection  on  that  plane  will 
be  parallel  and  equal  to  the  line  itself. 

Regarding  MI  as  the  given  line,  the  projections  of  all  its  points 
fall  together  at  in\  that  is  to  saj,  if  a  line  be  perpendicular  to  a 
plane,  its  projection  on  that  plane  is  a  point ;  which  point  lies  in  the 
line  itself. 


If  a  line  be  inclined  to  a  plane,  Us  projection  on  that  plane  will  be 
shorter  than  the  line  itself;  thus,  MN^  being  the  hypothenuse  of 
the  right-angled  triangle -JZ/iV^,  is  greater  than  the  base  /iV,  or  its 
equal  nin. 

1 5 .  If  two  lines  are  parallel  in  space,  their  projections  on  the  same 
plane  will  also  be  parallel.  In  Fig.  7  let  DE  be  parallel  to  MN\ 
then  its  projecting  plane  De  is  parallel  to  the  projecting  plane  Mn. 
The  plane  XY  cuts  these  two  parallel  planes  in  the  lines  m?i,  de^ 
which  are  therefore  parallel  to  each  other. 

Prolong  J/iYto  any  point  L  on  the  opposite  side  of  XY^  of 
which  I  is  the  projection ;  also  produce  Mm  and  draw  IK  perpen- 
dicular to  it.  Then  mKr=  11^  and  ml  =  nX,  whence  MI  —  Mm 
—  iW^,  and  MK—  Mm  -\-  II.  Xow  MX  is  the  hypothenuse  of 
the  triangle  MIX^  whose  base  IX  is  equal  to  ma.,  and  MI  is  the 
hypothenuse  of  the  triangle  MKI^  whose  base  is  equal  to  ml.  We 
see,  then,  that  the  true  length  of  a  line  in  space  is  equal  to  the  hy- 
pothenuse of  a  right-angled  triangle,  whose  base  is  equal  to  the  pro- 
jection of  the  line  on  any  plane ;  the  altitude  being  equal  to  the 
diiference  of  the  projecting  perpendiculars  of  the  two  extremities  of 


8 


DESCRIPTIVE   GEOMETRY. 


tlie  line  if  tliey  lie  on  tlie  same  side  of  the  plane,  and  equal  to  their 
sum  if  tbey  lie  on  opposite  sides. 

16.  If  two  lines  intersect  in  space,  their  projections  upon  any 
plane  will  either  intersect  each  other,  or  thej  will  coincide.  If  the 
plane  of  the  given  lines  is  perpendicular  to  the  given  plane,  it  will 
be  their  common  23rojecting  plane,  and  the  projections  will  coincide ; 
thus  7)in  is  the  projection  of  both  MJV  and  IW.  But  the  lines 
FC^GII^  which  intersect  at  7?,  have  not  a  common  projecting 
plane ;  and  the  two  planes  Gh^  Fc^  intersect  in  a  line  Ri\  wdiich 
must  be  perpendicular  to  XY^  and  is  therefore  the  project- 
ing line  of  R ;  that  is  to  say,  the  intersection  of  the  projections  of 
two  intersecting  lines  upon  the  same  plane  is  the  projection  of  the 
intersection. 

But  the  projections  may  intersect  although  the  lines  themselves 
do  not ;  it  is  evident  that  in  the  plane  Hg  many  lines  may  be  drawn 
which  would  pass  either  under  or  over  FC^  and  in  the  jDlane  Cf 
many  others  which  would  not  intersect  Gil. 

17.  A  line,  like  a  point,  is  represented  by  its  projections  on 
H  and  T;   thus  in  Fig.  8,  cd  is  the  horizontal,  and  c'd'  is  the  verti- 


cal, projection  of  the  line  CD.  This  is  shown  pictorially  in  Fig.  9, 
where  Cd  is  the  horizontal  projecting  plane,  and  Cd'  is  the  vertical 
projecting  plane,  of  the  given  line.  In  general,  a  nne  is  fully  de- 
termined by  its  projections ;  for  if  they  are  given  the  projecting 
planes  can  be  constructed,  and  since  the  line  must  lie  in  each  of 
them,  it  will  be  their  intersection,  if  they  have  one.  The  limited 
line  CD  is  a  portion  of  the  line  X  Y  of  indefinite  length,  repre- 
sented by  indefinitely  extending  the  projections,  as  xy,  x'y'. 

18.  In  general,  then,  any  indefinite  line  a?y  in  H  may  be  assumed 


DESCRIPTIVE    GEOMETRY.  9 

as  the  liorizontal  projection,  and  any  indefinite  line  x'y'  in  T  as  tlie 
vertical  projection,  of  a  line  whose  position  in  space  is  thus  defi- 
nitely fixed. 

This,  however,  is  subject  to  the  restriction  that  if  either  pro- 
jection be  perpendicular  to  AB,  the  other  projection,  whether  it  be 
a  point  or  a  right  line,  must  lie  in  the  prolongation  of  that  per- 
pendicular. 

Thus  in  Fig.  8,  the  horizontal  projection  Teg  is  perpendicular  to 
AB;  but  (10)  the  vertical  and  the  horizontal  projections  of  any 
point  must  both  lie  in  the  same  perpendicular  to  the  ground  line, 
consequently  h'g'  will  lie  in  lig  produced. 

In  this  case  the  indefinite  projections  on  H  and  Y  do  not  suffice 
to  determine  the  line ;  the  projecting  planes  coincide,  being  perpen- 
dicular to  A B  at  the  same  point,  and  have  no  line  of  intersection. 
If,  as  in  Fig.  8,  the  j)rojections  of  two  points  of  the  Kne  are  dis- 
tinguished by  letters,  the  line  is  determined,  but  such  a  representa- 
tion is  most  unsatisfactory  and  difficult  to  read :  the  line  lies  in  a 
profile  plane,  and  its  projection  thereon  should  always  be  added. 

19.  A  line  in  space  may  lie  in  either  plane  of  projection;  its 
projection  on  the  other  then  falls  in  the  ground  line.  If  it  lies  in 
neither  plane,  it  may  be  parallel  to  one  only,  parallel  to  both,  or 
inclined  to  both. 

If  a  line  is  perpendicular  to  one  plane,  it  is  parallel  to  the  other ; 
its  projection  on  the  first  is  a  point,  and  its  projection  on  the  other 
is  perpendicular  to  AB. 

If  the  line  is  parallel  to  one  plane  and  inclined  to  the  other,  its 
projection  on  the  first  is  parallel  to  the  line  itself,  and  its  projec- 
tion on  the  other  is  parallel  to  AB. 

If  the  line  is  parallel  to  both  planes,  the  line  itself  and  both  its 
projections  are  parallel  to  AB. 

20.  The  doubly  oblique  positions  may  be  divided  into  two 
groups ;  one  including  those  which  ascend  as  they  recede,  like  the 
ones  drawn  on  the  sloping  plane  J/Z,  Fig.  10;  and  the  other 
including  those  which  descend  as  they  recede,  like  the  ones  on  the 
farther  plane  J/iT. 

Those  of  the  first  group  may  cross  the  first  angle,  piercing  the 
front  part  of  H  and  the  upper  part  of  V ;  they  may  cross  the  third 


10 


DESCRIPTIVE    GEOMETRY. 


angle,  piercing  tlie  lower  part  of  V  and  the  rear  part  of  H ;  or  tliey 
may  intersect  AB,  in  wliicli  case  tliej  lie  wholly  in  the  second  and 
fourth  angles. 

Those  of  the  second  group  may  cross  the  second  angle,  piercing 


Fig.  10 


the  upper  part  of  V  and  the  rear  part  of  H ;  they  may  cross  the 
fourth  angle,  piercing  the  front  part  of  H  and  the  lower  part  of  V ; 
or  they  may  intersect  AB,  in  which  case  they  lie  wholly  in  the  first 
and  third  angles. 

Lines  of  either  group,  as  shown  in  Fig.  10,  may  inchne  either  to 
the  right  or  to  the  left,  in  which  case  their  projections  will  be 
inclined  to  AB;  or  they  may  do  neither;  lying  then  in  profile 
planes,  their  projections  on  H  and  Y  are  perpendicular  to  AB. 

21.  In  Fig.  11  the  projections  on  H  and  V  consist  merely  of 


d'  g^e 


B            C 

2 

!                     '            1 

d         c 

4 

9 

e 
12 

3 

a 

E 

Fig. 

Fig.  n 


two  lines  below  AB  and  perpendicular  to  it,  and  two  ]3oints  above 
AB.  These  representations  being  identical,  it  would  be  impossible 
to  distinguish  between  them,  or  to  decide  with  certainty  what  either 
was  intended  to  show,  were  they  not  lettered.  By  the  aid  of  the 
letters  we  perceive  that  CD  is  a  line  of  limited  length,  perpendicu- 
lar to  V,  and  lying  in  the  first  angle,  while  OE  is  a  limited  vertical 
line  situated  in  the  third  angle.     These  things  are  seen  by  a  single 


DESCRIPTIVE   GEOMETRY. 


11 


glance  at  Fig.  12;   which  illustrates  the  value  of  the  profile,  some- 
times in  even  very  simple  cases.. 

22.  In  Fig.  13  are  given  the  projections  of  a  horizontal  line 
lying,  as  pictorially  represented  in  Fig.  14,  in  the  second  angle. 
Being  horizontal,  its  vertical  projecting  plane  is  also  horizontal,  and 
therefore  cut  by  T  in  a  horizontal  line ;  that  is  to  say,  the  vertical 
projection  c'd'  is  parallel  to  AB  (19). 


.  Eiaae 


Fig.  15  gives  the  projections  of  an  inclined  line,  in  the  first  angle 
and  parallel  to  V,  as  seen  in  Fig.  16.  Its  horizontal  projecting  plane 
is  therefore  parallel  to  V,  and  since  these  two  parallel  planes  are  cut 
by  H  in  parallel  lines,  the  horizontal  projection  cd  is  parallel  to 
AB  (19). 

23.  The  doubly  oblique  lines,  not  being  parallel  to  either  plane, 
pierce  them  both.  In  relation  to  these,  beginners  sometimes  find 
difficulty  in  reading  the  drawings, — that  is,  in  forming  by  aid  of 
the  projections  alone,  clear  perceptions  of  the  actual  positions  of  the 
lines  in  space.     This  difficulty  may  perhaps  be  lessened  by  consider- 


12 


DESCKIPTIYE    GEOMETRY. 


y\^      Fig.  23 


Fig.  24 


DESCRIPTIVE   GEOMETRY.  13 

ing  at  first  such  as  do  not  meet  the  ground  line,  and  confining  the 
attention  to  the  portions  intercepted  between  H  and  V. 

In  Fig.  17  is  given  a  pictorial  representation  of  a  line  which 
crosses  the  first  angle.  In  order  to  draw  the  projections  of  such  a 
line  we  may  assume  <?,  Fig.  18,  as  the  horizontal  projection  of  any 
point  lying  in  H  in  front  of  V ;  its  vertical  projection  c'  will  lie  in  AB  : 
also  assume  d '  as  the  vertical  projection  of  a  point  in  V  above  H  ;  its 
horizontal  projection  d  must  also  lie  in  AB ;  therefore  cd  is  the  hori- 
zontal and  c'd'  is  the  vertical  projection  of  the  required  line. 

24.  So,  by  assuming  the  projections  of  a  point  in  each  plane, 
it  is  easy  to  represent  a  line  crossing  any  angle  at  pleasure.  Thu& 
Fig.  19  shows  one  which  crosses  the  second  angle;  its  representa- 
tion in  projection.  Fig.  20,  differs  from  Fig.  18  only  in  this,  that  g 
lies  above  instead  of  below  AB,  being  the  horizontal  projection  of  a 
point  in  H  behind  T. 

Completing  the  series,  the  four  following  figures  represent,  pic- 
torially  and  in  projection,  lines  crossing  the  third  and  the  fourth 
angles.  And  it  is  observed  that  the  two  projections  of  the  inter- 
cepts in  the  first  angle.  Fig.  18,  and  in  the  third  angle.  Fig.  22, 
do  not  intersect  each  other.  If  the  intercept  lies  in  either  of  the 
other  angles,  its  projections  cross  each  other,  the  intersection  being 
above  AB  if  it  lies  in  the  second,  as  in  Fig.  20,  and  helow  AB,  as  in 
Fig.  24,  if  it  lies  in  the  fourth. 

25.  To  find  the  traces  of  a  line  whose  projections  are  given. 
The  points  in  which  a  line  pierces  H  and  T  are  called  respectively 
its  horizontal  trace  and  its  vertical  trace.  In  constructing  Fig.  18, 
as  explained  in  (23),  the  traces  w^ere  assumed,  and  determined  the 
projections  of  the  line ;  by  merely  reversing  the  j^rocess,  the  traces' 
may  be  found  if  the  projections  are  given.  For  if  the  line  in  space 
be  produced,  its  projections  w411  be  extended;  the  distance  of  any 
point  of  the  horizontal  projection  from  AB  shows  the  distance  of  the 
corresponding  point  in  the  line  from  V,  which  becomes  zero  when 
that  projection  meets  AB ;  and  similarly  the  altitude,  or  distance 
from  H,  becomes  zero  when  the  vertical  projection  meets  AE. 
Therefore,  if  in  Fig.  18  the  projections  mn^  Tn'n'  are  given,  pro- 
duce mn  to  cut  AB  in  d\  this  will  be  the  horizontal  projection  of 
the  vertical  trace :  the  other  projection  must  lie  on  a  perpendicu- 


14 


DESCRIPTIVE    GEOMETRY. 


lar  to  AB  at  d^  and  also  on  tlie  prolongation  of  the  vertical  projec- 
tion m'li' ;  therefore  it  is  at  their  intersection  d' .  Prodnce  vin'  to 
cut  AB  at  g' ^  the  vertical  projection  of  the  horizontal  trace ;  at  c' 
draw  a  perpendicular  to  AB,  cutting  mn  produced  in  <?,  the  other 
projection.      TJie  points  C  and  D  are  the  traces  sought. 

26.  In  attempting  to  find  the  traces  of  a  line  given  in  this  man- 
ner, it  may  happen  that  both  its  projections  meet  the  ground  line  at 
the  same  point.  This  means  simply  that  the  line  pierces  both  H  and 
V  at  that  point,  and  therefore  intersects  tlie  ground  line.  Tliis  is  the 
case  in  Fig.  25 ;  and  as  the  line  JlfiTlies  in  the  first  angle,  it  must 
when  prolonged  pass  into  the  third  angle  after  crossing  AB,  as  more 
distinctly  seen  in  the  profile,  Fig.  26,  the  addition  of  which  third 


riG.27 


projection  greatly  facilitates  the  reading  in  such  cases.  In  Fig.  27 
the  given  limited  portion  MN  of  the  line  lies  in  the  second  angle ; 
when  prolonged  the  line  cuts  AB  at  C^  and,  as  shown  in  the  profile, 
Fig,  28,  passes  on  into  the  fourth  angle. 

27.  OMique  Lines  with  Coincident  Projections.  In  Fig.  29  tlie 
point  M  is  as  far  behind  V  as  it  is  above  H,  and  in  consequence  m 
and  m!  fall  together.  The  point  D  is  as  far  below  H  as  it  is  in  front 
of  V,  whence  d  and  d'  also  fall  together.  -  Therefore  the  vertical 
and  the  liorizontal  projections  of  the  line  MD  coincide  in  one  line. 


DESCRIPTIVE    GEOMETRY. 


15 


Every  other  point  of  tlie  line  will  therefore  be  represented  by  coin- 
cident projections,  as  ?i,  71' ;  all  such  points  lie  in  either  the  second 
angle  or  the  fourth,  and  since  they  are  equidistant  from  H  and  V, 
they  and  the  line  itself  lie,  as  shown  in  the  profile,  Fig.  30,  in  a 
plane  which  bisects  those  two  angles  (13). 


Fig.  29 


JFiaSS 


Fig.  30 


d' 

e 
n' 

e' 

iW 

c' 

n                c 

) 

m            g 

g' 

c 

Fig.  31 

d 

n'     A 

H^- 

my 

Gy\             : 

/ 

X    m 

n/ 

> 

' 

Fig.  32 

.iV 


Fig.  34 


28.  Oblique  Lines  in  Profile  Planes.  In  Fig.  31  both  projections 
are  perpendicular  to  the  ground  line.  In  tins  case  these  projec- 
tions upon  the  principal  planes  are  utterly  inadequate  to  convey  a 
clear  idea  of  the  position  of  a  limited  portion  of  the  line,  even  with 
the  aid  of  the  letters.  The  indefinite  projections,  even  supposing  it 
to  be  known  which  is  the  vertical  and  which  the  horizontal,  do  not 
suffice  (18)  to  locate  the  line  in  space,  and  since  their  prolongations 
coincide,  it  is  impossible  by  their  use  to  find  the  traces.  The  obvious 
and  the  only  sensible  course  is  to  make  an  independent  and  detached 
drawing  in  profile,  as  shown  in  Fig.  32,  which  exhibits  in  the  clear- 
est possible  manner  the  position  of  the  line  in  relation  to  both  H 


10  DESCKIPTIVE   GEOMETRY. 

and  V,  whether  it  crosses  one  angle  or  another,  like  MN^  or  like 
GE  intersects  the  ground  line. 

29.  Lines  Parallel  to  Both  Principal  Planes.  A  line  which  is 
parallel  to  botli  H  and  V  is  parallel  to  the  ground  line.  It  may  lie 
in  one  of  those  planes;  where.it  is  its  own  projection,  the  projec- 
tion on  the  otlier  plane  falling  in  AB,     If  it  does  not  lie  in  either 

11  or  V,  both  its  projections  are  parallel  to  the  ground  line.  The 
projections  of  such  a  line  upon  the  principal  planes  are  sufficient  to 
locate  the  line  in  space,  and,  as  shown  in  Fig.  33,  they  suffice  to 
represent  it.  Nevertheless,  in  this  case  also  the  reading  of  the 
drawing  is  facilitated,  and  the  locution  of  the  line  more  clearly  indi- 
cated, as  shown  in  Fig.  34,  by  adding  a  profile. 

REPRESENTATION    OF    THE    PLANE. 

30.  The  intersection  of  a  plane  with  V  is  called  its  yertical  trace ; 
its  intersection  with  H  is  called  its  horizontal  trace ;  and  the  plane  i& 
represented  by  drawing  these  traces. 

Any  horizontal  plane,  being  parallel  to  H,  has  no  horizontal 
trace,  and  its  vertical  trace  is  parallel  to  AB.  Example :  the  ver- 
tical projecting  plane  of  6ZZ>,  Fig.  14. 

If  a  plane  is  parallel  to  V  it  has  no  vertical  trace,  and  its  hori- 
zontal trace  is  parallel  to  AB.  Example  :  the  horizontal  projecting 
plane  of  CD,  Fig.  16. 

If  a  plane  is  parallel  to  AB  and  inclined  to  H  and  V,  both  traces 
will  be  parallel  to  AB ;  or,  they  may  coincide  in  the  ground  line 
itself.     These  cases  will  be  illustrated  farther  on. 

If  a  plane  is  perpendicular  to  AB,  i.e.,  if  it  is  a  profile  plane, 
both  traces  are  perpendicular  to  AB.  Example :  the  projecting 
planes  of  KG,  Fig.  9. 

It  will  be  perceived  from  the  examples  here  quoted  that  the 
projections  of  all  lines  are  the  traces  of  their  projecting  planes  (14). 

31.  If  a  plane  be  inclined  to  AB,  it  will  cut  it  in  a  point;  the 
traces  must  intersect  at  this  point,  and  one  or  both  of  them  will  be 
incHned  to  AB.  Thus  in  Fig.  35,  the  oblique  plane  Jf Aleuts  AB 
at  D ;  its  horizontal  trace  is  dDc,  and  d'Dc'  is  its  vertical  trace. 

Such  a  plane  is  represented  in  projection  as  in  Fig.  36 ;  it  is 
designated  and  referred  to  as  the  plane  dDd' .     If,  as  often  hap- 


DESCRIPTIVE   GEOMETRY. 


17 


pens,  'itteution  is  to  be  confined  to  tliat  portion  of  the  plane  which 
lies  in  the  first  angle,  between  H  and  V,  the  parts  Dc^  Dc'  of  the 
traces  are  omitted ;  but  it  must  be  kept  in  mind  that  the  plane  is 
capable  of  indefinite  extension,  and  both  traces  can  be  indefinitely 
produced. 

It  is  to  be  observed,  in  regard  to  this  notation,  that  d  and  d'  are 
not  used  to  indicate  the  two  projections  of  the  same  point ;  d  merely 
designates  a  point  in  H,  and  d'  designates  a  point  in  V.  If  the  loca- 
tion in  space  of  a  particular  point  in  either  trace  is  to  be  indicated. 


Fig.  36 


the  two  projections  of  that  point  are  lettered  in  the  usual  manner ; 
thus  c,  o'  are  the  projections  of  a  point  in  Dd\  and  ^,  r'  those  of 
one  in  Dd ;  these  points  are  referred  to  as  0  and  R  respectively. 

32.  For  the  purpose  of  aiding  those  who  may  at  first  find  dif- 
ficulty in  reading  the  diagram,  Fig.  36,  there  are  placed  above  it 
drawings,  on  a  reduced  scale,  of  the  two  cards  with  their  slots 
which  form  the  planes  of  projection  in  the  model.  Fig.  35. 

In  looking  at  the  card  F",  the  paper  is  held,  or  supposed  to  be 
held,  in  a  vertical  position ;  while  in  looking  at  the  card  H^  it  is 
held,  or  imagined  to  be,  in  a  horizontal  position,  and  viewed  from 
above.  The  diagram  represents  both  these  cards  in  skeleton ;  make 
therefore  the  same  suppositions  in  regard  to  the  position  of  the 
paper  and  the  direction  in  which  it  is  to  be  viewed,  closing  the 
mental  eye  to  one  projection  while  studying  the  other.  By  persist- 
ent eft'orts  of  this  kind,  the  power  may  gradually  be  acquired  of 


18 


DESCRIPTIVE    GEOMETRY. 


reading  tlie  diagrams  with  ease — that  is  to  saj,  of  forming  by  tlie 
aid  of  tlie  projections  alone,  clear  mental  images  of  the  positions  and 
relations  of  the  lines  and  planes  which  thej  represent,  so  that  they 
will,  as  one  may  say,  stand  out  in  relief  with  stereoscopic  dis- 
tinctness. 

33.  In  regard  to  these  two  cards,  it  is  evident  that  the  direc- 
tions of  the  slots  are  entirely  arbitrary  and  independent  of  each 
otlier ;  but  when  put  together,  the  point  D  on  one  must  coincide 
with  tlie  point  D  on  the  other.  Which  is  only  another  way  of  say- 
ing that  from  the  same  point  on  AB  we  may  draw  one  line  in  any 
direction  on  H,  and  another  in  any  direction  on  V,  and  these  two 
lines  will  determine  a  plane,  of  whicli  they  are  the  traces. 

If  the  vertical  trace  is  perpendicular  to  AB,  the  plane  is  verti- 
cal, but  may  make  any  angle  with  V,  as  in  the  swinging  of  a  com- 
mon door  upon  its  hinges.  If  tlie  horizontal  trace  is  perpendicular 
to  AB,  the  plane  is  perpendicular  to  V,  but  may  make  any  angle 
with  H ;  as  in  the  opening  of  a  trap-door  whose  hinges  are  perpen- 
dicular to  the  wall.  The  plane  dDcV  illustrates  the  former,  and 
the  plane  tTif  illustrates  the  latter,  of  these  two  cases,  in  Figs.  37 
and  38. 


.Fig.  37 


^ 

y 

6! 

/ 

D 

' 

T 

/ 

t 

\^, 

>  Pig;  38 


34.  In  relation  to  the  angle  included  between  the  parts  of  the 
traces  in  front  of  V  and  above  H,  it  is  apparent  that  the  angle  pic- 
torially  represented  by  dDd'  in  Fig.  35  is  in  fact  acute,  while  in 
Fig.  37  dDd'  and  tTt'  represent  right  angles.     In  Fig.  39  dDd' 


DESCRIPTIVE   GEOMETRY. 


19 


represents  an  obtuse  angle ;  and  in  the  diagram,  Fig.  40,  are  given 
the  traces  of  the  same  plane,  similarly  lettered.  Above  are  added 
the  small  drawings  of  the  cards  with  their  slots,  for  forming  the 
planes  H  and  V  of  the  model. 

These  slots  are  inclined  to  AB  in  the  same  direction,  though 
the  angles  are  somewhat  different  in  the  two  cards.  A  moment's 
reflection  will  show  that  these  angles  might  be  made  exactly  the 
same,  and  also  that  if  they  were,  the  two  traces  dD,  Dd\  instead 
of  forming  an  angle  with  each  other  in  the  diagram,  would  form 


A 

-^i^"'  "" 

R 

^  ! 

A 

Di>\         " 

R 

/ 

y 

Fig.  39 


Fig.  40 


one  continuous  right  line ;  which  is  the  case  with  the  traces  of  the 
plane  tTt\  Clearly,  the  position  of  the  plane  shown  in  Fig.  39 
would  be  but  slightly  changed  by  this  modification. 

35.  In  Fig.  41,  the  plane  TT  is  parallel  to  AB;  consequently 
its  horizontal  trace  tt  and  its  vertical  trace  ff  are  both  parallel  to 
the  ground  line.  The  plane  DD  passes  through  AB,  which  there- 
fore constitutes  both  traces. 

The  diagram,  representing  these  two  planes  is  given  in  Fig.  42 ; 
but  it  is  very  obvious  that  they  are  represented  much  more  clearly 
in  the  profile,  Fig.  43. 

Since  any  number  of  planes  may  be  passed  through  the  ground 
line,  the  position  of  any  one  of  them  must  be  determined  by  some 
other  condition ;  but  when  it  is  determined,  its  true  relation  to  H 
and  V  is  at  once  shown  by  the  detached  profile. 

In  Fig.  44,  the  horizontal  trace  dd  coincides  with  the  vertical 
trace  d'd',  the  former  being  as  far  behind  V  as  the  latter  is  above 


20 


DESCinrnVE    GEOMETRY. 


H.  Again,  tt  is  the  liorizontal  trace  of  a  plane,  and  lies  as  far  in 
front  of  V  as  the  vertical  trace  t't'  is  below  H,  so  that  these  two 
traces  are  also  represented  bj  one  line.  Finally,  mn^  nn'n'  are  the 
projections  of  a  line,  parallel  to  AB,  in  the  fourth  angle,  and  equi- 
distant from  the  principal  planes.      The  superiority  of  the  profile, 


B 

y^ 

'  / 

t' 

r 

^ 

d' 

d' 

^ 

A 

^ 

d 

B 

t 

t 

Pig.  42 

d' 


n'         t' 


Fig.  44 


Fig.  43 


M* 


Fig.  45 


Fig.  45,  in  respect  to  distinctness  and  ease  of  comprehension,  is  too 
obvious  to  require  comment. 


GEOMETRICAL    PRINCIPLES    AND    DEDUCTIONS. 

36.  If  two  planes  intersect,  any  line  of  either  will  pierce  the 
other  in  a  point  of  their  common  line,  if  at  all ;  hence — 

1.  If  a  line  lie  in  a  plane,  the  traces  of  the  line  will  be  points 
in  the  corresponding  traces  of  the  plane. 

2.  To  draw  a  line  in  a  given  plane,  join  any  point  in  one  trace 
w-ith  any  point  in  the  other. 

3.  To  draw  a  plane  containing  a  given  line,  join  the  traces  of 
the  line  with  any  point  on  AB. 

Any  horizontal  line  in  a  given  plane  is  parallel  to  the  horizontal 
trace,  pierces  V  in  a  point  of  the  vertical  trace,  and  its  vertical  pro. 
jection  is  parallel  to  AB. 


DESCRIPTIVE    GEOMETRY. 


21 


If  a  line  in  a-  given  plane  is  parallel  to  T,  it  is  parallel  to  the 
vertical  trace,  pierces  H  in  a  point  of  the  horizontal  trace,  and  its 
horizontal  projection  is  parallel  to  AB. 

If  a  plane  contain  any  two  lines,  it  will  also  contain  any  third 
line  which  cnts  those  two. 

3  7.  In  illustration  of  the  above :  Let  it  be  required  to  draw  a 
line  in  the  plane  tTt\  Fig.  46.  Assume  c  as  the  horizontal  projec- 
tion of  a  point  in  the  horizontal  trace ;  its  vertical  projection  is  o' 
in  the  ground  line.  Let  d'  be  the  vertical  projection  of  a  point  in 
the  other  trace,  then  its  horizontal  projection  is  d  in  the  ground 
line ;  cd^  c'd\  are  the  projections  of  a  line  which  lies  in  the  giveu 
plane.  It  follows  from  this^  that  if  one  projection  of  a  point  in  a 
given  plane  be  assumed,  the  other  can  be  found  by  drawing  through 
the  assumed  one,  the  corresponding  projection  of  a  line  in  the  plane. 
Then  the  otlier  projection  of  the  point  must  lie  on  the  other  pro- 
jection of  the  line.  For  example,  suppose  the  horizontal  projection 
o  in  Fig.  46  to  have  been  assumed.  Join  o  with  any  point  c  of  the 
horizontal  trace,  and  produce  this  horizontal  projection  to  cut  AB 


in  d.  Since  this  line  is  to  lie  in  the  plane,  its  vertical  projection  is 
cd\  upon  which  must  lie  the  vertical  projection  o' ,  The  point  O 
thus  determined  lies  in  the  given  plane. 

Again,  let  it  be  required  to  draw  in  the  plane  tTt' ^  Fig.  47,  a 
horizontal  line  at  a  given  distance  above  H. 

Draw  c'd'  parallel  to  AB,  at  the  given  distance  above  it:  this  is 
the  vertical  projection  of  the  required  line,  and  d'  that  of  its  verti- 
cal trace,  which  is  horizontally  projected  at  d  in  AB.  Therefore 
dc  parallel  to  Tt  is  the  horizontal  projection  of  the  required  line. 


22 


DESCRIPTIVE   GEOMETRY. 


Let  it  be  further  required  to  draw  in  the  same  plane  a  line 
parallel  to  V,  at  a  given  distance  in  front  of  it.  The  horizontal 
projection  is  mn^  parallel  to  AB  and  at  the  given  distance  below  it ; 
the  line  pierces  H  at  the  point  iT,  whose  vertical  projection  is  n'  on 
AB,  and  n'm'  parallel  to  Tt'  is  the  vertical  projection  of  the  re- 
quired line. 

38.  The  two  lines  CD  and  JfiT evidently  intersect;  and  since 
they  cannot  intersect  in  more  than  one  point,  the  test  of  the  accu- 
racy of  the  constructions  lies  in  this,  that  the  intersection  o  of  the 
horizontal  projectioils,  and  the  intersection  o'  of  the  vertical  pro- 
jections, lie  on  the  same  perpendicular  to  the  ground  line. 

39.  To  draw  two  lines  which  shall  intersect.  This  may  be  done 
by  assuming  the  point  of  intersection  6',  Fig.  48 ;  c  and  c  must 
necessarily  lie  on  the  same  perpendicular  to  AB  (10).  The  hori- 
zontal projection  of  each  line  must  pass  through  c,  and  its  vertical 
projection  through  c' ;  but  the  directions  of  mc?i,  in'e'n^  as  well  as 
those  of  jpcr^  jp'c'r\  are  entirely  arbitrary,  with  the  excej^tion  that 
if  one  projection  of  either  line  is  perpendicular  to  AB,  the  other 
projection  of  that  line  must  be  so  likewise  (18). 

The  two  lines  GL^  DE^  in  Fig.  48,  intersect  at  0\  the  hori- 
zontal projections  intersect  at  6>,  but  the  vertical  projections  coin- 
cide.    This  merely  shows  that  the  plane  determined  by  the  two 


Fig.  49 


lines  is  perpendicular  to  V,  and  is  their  common  vertical  projecting 
plane  (16). 

40.  To  draw  two  lines  which  shall  not  lie  in  the  same  plane. 

^Neither  the  vertical  nor  the  liorizontal  projections  can  coincide, 
since  if  they  did  the  two  lines  would  have  a  common  projecting 


DESCRIPTIVE    GEOMETRY. 


23 


plane.  The  horizontal  projections  must  therefore  cross  eac-h  other, 
and  so  must  the  vertical  projections ;  but  these  two  points  of  inter- 
section must  not  lie  in  the  same  perpendicular  to  AB. 

Thus  in  Fig.  49,  the  horizontal  projections  of  the  lines  CG^ 
LO^  intersect;  let  this  point  of  intersection  be  the  horizontal  pro- 
jection m  of  a  point  M  upon  L  0^  then  its  vertical  projection  is  m' 
upon  Vo'.  Let  the  same  intersection  be  the  horizontal  projection  d 
of  a  point  upon  CG^  tlien  its  vertical  projection  is  d'  upon  eg'. 
Similarly,  the  intersection  of  I'o'  and  c'g'  is  the  common  vertical 
projection  of  two  points,  J^  upon  L  0^  and  E  upon  CG. 

41.  If  two  parallel  planes  are  cut  by  a  third  plane,  the  lines  of 
intersection  are  parallel.  Therefore,  in  order  to  represent  a  plane 
parallel  to  one  of  which  the  traces  are  given,  draw  the  vertical 
trace  of  the  second  parallel  to  that  of  the  first,  and  the  horizontal 
trace  of  tlie  second  parallel  to  the  horizontal  trace  of  the  first.  If 
the  new  plane  is  required  to  be  so  located  as  to  satisfy  some  special 
condition,  it  is  clear  that  the  determination  of  one  point  in  either 
trace  is  sufficient. 

For  example :  Let  it  be  required  to  draw  a  plane  parallel  to  the 
given  plane  tTt\  Fig.  50,  through  the  given  point  0.  Draw, 
through  the  given  point,  a  line  parallel  to  the  horizontal  trace ;  its 


Horizontal  projection  passes  through  o  and  is  parallel  to^  Tt,  its 
vertical  projection  passes  througli  o'  and  is  parallel  to  AB.  This  is 
a  line  of  the  required  plane,  and  pierces  V  at  the  point  C^  whose 
vertical  projection  c  is  a  point  in  the  vertical  trace,  which  is  d'c' D 
parallel  to  Tt' .  This  trace  cuts  AB  at  />,  and  the  horizontal  trace 
Dd  is  parallel  to  Tt. 


24 


DESCRIPTIVE   GEOMETRY. 


42.  Let  ZJf,  LN^  Fig.  51,  be  two  planes,  and  Zi^  their  line 
of  intersection.  From  any  point  P  let  fall  upon  these  planes  the 
perpendiculars  PR^  P0\  these  two  lines  determine  a  plane  which 
is  perpendicular  to  both  the  others,  and  therefore  to  LF\  it  also 
cuts  them  in  the  lines  CRD^  COE^  which  meet  at  C  on  LF.  But 
LF  is  perpendicular  to  the  plane  OPR^  and  therefore  to  the  lines 
CEy  CD^  which  pass  through  its  foot  in  that  plane.  Now,  regard- 
ing ZJf  as  a  plane  of  projection,  andZT^as  the  trace  upon  it  of 
any  plane  LIi\  then  CR  is  the  projection  of  PO^  a  perpendicular 
to  LN^  and  the  trace  LF  is  perpendicular  to  the  projection  CR. 

43.  Therefore,  if  a  line  be  perpendicular  to  a  plane,  the  yertical 
projection  of  the  line  will  be  perpendicular  to  the  vertical  trace  of 
the  plane,  and  the  horizontal  projection  will  be  perpendicular  to  the 
horizontal  trace. 

And  conversely :  if  the  projections  of  the  line  are  respectively 
perpendicular  to  the  traces  of  the  plane,  the  line  itself  is  perpen- 
dicular to  the  plane. 

In  illustration,  let  it  be  required  to  draw  through  the  point  0, 
Fig.  52,  a  line  perpendicular  to  the  plane  tTt.   Since  the  projec- 


/    0 

p 

'1 

/ 

V"   y 

Fig.  52 


Fig.  53 


tions  of  the  line  must  pass  through  those  of  the  point,  we  have 
merely  to  draw  m'o'n'  perpendicular  to  Tt' ^  and  mon  perpendicular 
to  Tt ;  then  MN  is  the  required  line. 

Or,  having  given  the  line  MN  and  the  point  6^,  let  it  be  re- 
quired to  draw  a  plane  through  that  point  and  perpendicular  to  the 
line.  Here  again,  since  the  directions  of  the  traces  are  known,  the 
determination  of  one  point  in  either  trace  suffices  to  locate  the 
plane ;  and  this  is  effected  by  drawing  through  the  point  a  parallel 
to  the  other  trace.     Thus,  to  find  a  point  in  the  horizontal  trace,  draw 


DESCRIPTIVE    GEOMETRY.  25 

tlirongli  C  a  line  parallel  to  the  vertical  trace ;  its  vertical  projec- 
tion is  o'd'  perpendicular  to  mn  ^  its  horizontal  projection  is  cd 
parallel  to  AB,  and  it  pierces  ^in  the  point  d^  d' .  The  horizontal 
trace  of  the  required  plane  is  therefore  tdT  perpendicular  to  mn^ 
and  tlie  vertical  trace  is  Tt'  perpendicular  to  rrin' . 

44.  In  Fig.  53,  let  ZZ  be  a  plane  of  projection,  iTiV^  a  pro- 
jecting plane  perpendicular  to  it,  of  which  dd  is  the  trace,  andP(? 
a  line  perpendicular  to  iViVand  consequently  parallel  to  LL.  Then 
po^  the  projection  of  PO^  is  parallel  to  that  line  itself,  and  therefore 
perpendicular  to  NN  and  to  its  trace  dd.  Now  P  (9,  being  per- 
pendicular to  ^iY,  is  perpendicular  to  all  right  lines  drawn  through 
its  foot  in  that  plane,  as  OR^  0S\  and  the  projections  of  all  these 
lines  fall  in  the  trace  dd^  which  is  perpendicular  to  po\  conse- 
quently we  have,  that  the  projection  of  a  right  angle  will  be  a  right 
angle,  if  one  of  its  sides  is  parallel  to  the  plane  of  projection. 

REVOLUTION    AND    COUNTER-KEVOLUTION. 

45 .  The  axis  of  a  circle  is  a  right  line  passing  through  its  centre, 
and  perj)endicular  to  its  plane.  A  point  is  said  to  revolye  about  a 
right  line  as  an  axis,  when  it  describes  the  circumference  of  a  circle 
whoso  centre  is  in  the  axis,  and  whose  plane  is  perpendicular  to  the 
axis. 

When  all  the  points  of  any  geometrical  magnitude  move  in  this 
manner,  without  change  of  relative  position  and  therefore  with  tlie 
same  angular  velocity,  the  whole  magnitude  is  said  to  revolve  about 
the  right  line  as  an  axis. 

It  will  be  found  in  subsequent  operations,  that  magnitudes 
under  consideration  can  often  be  thus  revolved  into  positions  in 
which  certain  processes  can  be  more  conveniently  executed ;  after 
which  they  are  revolved  back  again  into  their  original  positions; 
and  this  restoration  is  called  counter-revolution. 

46.  One  of  the  simplest- examples  of  this  kind  of  manipulation 
is  the  following :  Given ^  a  line  lying  in  a  plane  of  projection,  and 
a  point  not  in  the  plane ;  Pequired^  to  show  where  the  point  will 
fall,  when  revolved  about  the  line  into  the  plane. 

This  is  pictorially  illustrated  in  Fig,  54,  where  EF  is  the  line, 
lying  in  the  plane  LL^  and  P  the  given  point.     The  plane  of  rota- 


26 


DESCRIPTIVE  Geometry, 


tion  is  WW^  perpendicular  to  EF  and  therefore  to  ZL ;  it  contains 
the  projecting  line  P/?  of  the  given  point,  and^  lies  in  the  trace 
dd^  which  is  perpendicular  to  EF  and  cuts  it  at  C,  the  centre  of 
the  circular  patli  of  P. 

Consequently  tlie  point  will  fall  in  the  plane  LL  either  at  r  or 


Fig.  55 


at  s  on  the  trace  dd^  at  a  distance  from  6^  equal  to  PC  i\\Q  radius 
of  the  circle.  And  it  is  seen  that  PC  is  the  hypothenuse  of  a  tri- 
angle, of  which  the  altitude  Pp  is  the  distance  of  the  point  from  the 
plane,  and  the  base  ^  6^  is  the  distance  of  the  projection  of  the  point 
from  the  axis.  The  axis  may  pass  through  this  projection ;  that  is, 
P  may  coincide  with  (7,  in  in  which  event  the  projecting  line  Pj)  is^ 
the  radius  of  the  circle. 

47.  The  construction  in  projection  is  shown  in  Fig.  55,  where 
0  is  the  given  point.  First :  to  revolve  this  point  about  CD  into 
the  horizontal  plane.  Draw  through  <?,  the  horizontal  projection 
of  the  point,  an  indefinite  perpendicular  to  cd^  cutting  it  in  m ;  the 
point  will  fall  somewhere  upon  this  line,  which  is  the  trace  of  the 
plane  of  rotation.  The  distance  of  the  point  from  the  plane  is  o'ic^ 
the  projecting  line,  and  om  is  the  distance  of  the  projection  from 
the  axis.  Set  off  xy  =  om,  then  o'y  is  the  true  distance  of  the 
point  from  m;  therefore  set  off  mo''  =  o'y,  and  o"  is  the  position 
of  0  after  revolution. 

Second :  to  revolve  0  about  PR  into  the  vertical  plane.  Draw 
through  o  an  indefinite  perpendicular  toj^V,  cutting  it  at  /?-,  and 
on  this  perpendicular  set  off  no'" ,  equal  to  the  hypothenuse  of  a 
triangle  whose  base  is  on,  and  whose  altitude  is  equal  to  xo,  the 


DESCRIPTIVE    GEOMETRY. 


27 


distance  of  0  from  V  :  this  triangle  may  be  conveniently  constructed 
by  setting  off  oiij)'?'',  ne  =  xo^  giving  o'e  as  the  true  distance  of  O 
from  n.  The  line  o'e  is  of  course  not  actually  to  be  drawn,  but  its 
length  being  taken  in  the  dividers,  is  set  off  from  n  as  no'" ^  and  o'" 
thus  located  is  tlie  required  position ;  similarly,  o'y  is  measured, 
but  not  drawn. 

48.  Since  a  right  line  and  a  point  outside  of  it  are  sufficient  to 
determine  a  plane,  the  point  0  in  Fig.  55  may  be  regarded  as  lying 
in  a  plane  of  w^hich  cd  is  the  horizontal  trace ;  and  when  this  point 
has  been  revolved  about  that  trace  into  the  position  o" ^  it  is  clear 
that  the  whole  plane  just  mentioned  will  coincide  with  H,  so  that 
any  lines  or  figures  which  it  had  contained  originally,  or  which  may 
now  be  drawn  upon  it,  will  be  seen  in  their  true  forms,  dimensions, 
and  relative  positions. 

In  like  manner,  the  same  point  O  lies  in  another  plane  of  which 
f'r'  is  the  vertical  trace,  and  when  0  reaches  o" ^  that  plane  coin- 
cides with  V.  x\ny  plane,  then,  can  be  revolved  about  either  trace 
into  the  corresponding  plane  of  projection ;  and  it  is  clear  that  by 
revolving  it  about  a  line  parallel  to  one  of  those  traces,  it  may  be 
made  parallel  to  the  corresponding  plane. 

49.  Counter-reYolution  of  a  Plane.  In  Fig.  ^^^  let  the  point/* 
be  revolved  about  the  axis  EF  into  the  j^lane  of  projection  LL  as 
in  Fig.  54,  ^''  being  its  revolved  position,  and  G  the  centre  of  its 


circular  path.     The  plane  PCEwqtn  coincides  with  LL\  assuming 
o"  to  be  the  revolved  position  of  any  point  therein  other  than  P, 


28  DESCRIPTIVE    GEOMETRY. 

it  is  required  to  find  its  position  wlien  the  plane  is  revolved  back 
again. 

Draw  ]p"o"  and  produce  it  to  cut  the  axis  in  D ;  then  the  re- 
quired original  position  of  o"  must  lie  on  the  line  PD^  whose  pro- 
jection on  LL  is  j?Z).  The  plane  of  rotation  of  the  second  point  is 
parallel  to  that  of  the  first,  and  its  trace  is  o"  G  perpendicular  to 
F.F\  which  cuts  jpD  in  6>,  the  projection  of  the  point  sought. 
Therefoi-e  erect  at  r?  a  j)erpendicular  to  LL^  cutting  PD  in  0^  the 
required  point. 

50.  This  construction,  in  projection,  is  shown  in  Fig.  57, 
where  EF  in  the  horizontal  plane  is  the  axis,  P  the  first  point  in 
its  original  position,  andj^''  its  revolved  position  determined  as  in 
Fig.  55.  Take  o"  as  the  revolved  position  of  tlie  second  point ; 
draw  j?"(>''  and  produce  it  to  cut  the  prolongation  Gi  fe  in  d.  This 
is  the  horizontal  projection  of  a  point  in  H,  therefore  its  vertical 
projection  is  d'  in  AB,  and  j9<^,  jp'd\  are  the  projections  of  the  line 
PD  whicli  contains  the  point  sought.  In  the  counter-revolution, 
o"  describes  a  circle  in  a  vertical  plane  perpendicular  to  the  axis,  of 
which  the  horizontal  trace  is  perpendicular  to  <?/",  and  cuts^^  in  6>, 
the  horizontal  projection  of  the  required  point;  o'  \v^  i^'d'  \^  its 
vertical  projection. 

SEPARATE    CONSTRL^CTION    OF    THE    VERTICAL    AND    HORIZONTAL 
PROJECTIONS. 

51.  The  combination  of  the  projections  on  H  and  V  in  one 
diagram,  and  the  use  of  AB  to  represent  one  plane  in  reading  tlie 
vertical  and  the  other  in  reading  the  horizontal  projection,  is  often 
a  source  of  perplexity  at  first,  even  to  those  thoroughly  familiar 
with  the  various  views  which  represent  solid  objects  in  ordinary 
mechanical  drawing.  Moreover,  this  combination  is  often  a  cause 
of  excessive  and  needless  obscurity ;  the  various  lines  used  in  repre- 
senting magnitudes,  operations  of  revolution,  counter-revolution, 
and  what  not,  upon  one  plane,  becoming  so  interwoven  with  those 
of  the  projections  upon  the  other,  as  to  present  a  bewildering  maze 
€ven  to  tlie  expert  in  reading  such  diagrams. 

Now,  the  fact  that  these  two  projections  can  be  and  always  have 
been  thus  combined  is  not  at  all  a  good  reason  why  they  always 


DESCRIPTIVE    GEOMETRY. 


29 


^lioiild  be.     Tliey  may  be  constructed  separately  as  sliown  in  Fig. 
58,  where  in'n'  is  the  vertical  projection  of  a  line,  i/7/ represent- 


3/ 


VERTICAL  PROJECTION 
OR  FRONT  ELEVATION 


H    H- 


PROFILE  OR 
END  ELEVATION 

.Fig.  58 


HORIZONTAL  PROJ. 

PLAN,  OR  TOP  VIEW  \m, 


ing  the  horizontal  plane;  mn  is  the  horizontal  projection,  YY 
representing  the  vertical  plane ;  and  at  the  left  of  the  vertical  pro- 
jection is  the  profile,  where  the  two  planes  are  shown  in  their  true 
relative  positions.  These  three  views  correspond  to  what  in  working 
drawings  are  called  the  front  elevation  or  side  view,  the  top  view 
or  plan,  and  the  end  view  or  end  elevation.  For  the  purpose  oi 
comparison,  the  combined  diagram  of  the  projections  of  the  same 
line  is  sliown  in  Fig.  59 ;  and  there  are  no  doubt  many  to  whom 
the  latter  will  seem  far  less  clear,  and  more  difficult  to  read,  even 
in  regard  to  so  simple  a  magnitude  as  this  oblique  line. 

In  what  follows,  both  methods  wdll  be  used,  as  circumstances 
may  indicate  one  or  the  other  to  be  the  more  convenient.  The 
student,  of  course,  may  use  either  in  any  case  at  pleasure.  It  is 
desirable  that  he  should  become  familiar  with  both,  but  in  any 
given  construction  he  alone  can  tell  which  method  seems  to  him 
the  clearer,  and  that  is  tlie  one  for  him  to  adopt. 


SUPPLEMENTARY    PLANES    OF    PROJECTION. 

52.  Thus  far  but  three  planes  of  projection  have  been  con- 
sidered, viz.,  H,  V,  and  the  profile  plane.  It  will,  however,  fre- 
quently be  found  convenient  to  make  use  of  others,  which  may  be 
called  supplementary  planes ;  upon  these  the  object  is  projected, 
remaining  fixed  in  respect  to  the  principal  planes. 

The  positions  of  such  supplementary  planes  are  determined 
wholly  by  conditions  of  convenience,  and  therefore  depend  upon 


30 


DESCKTPT[YE    GEOMETRY. 


the  nature  of  the  object;  but  thej  are  in  the  great  majority  of 
cases  such  that  the  j)lanes  are  j^erpendicular  to  one  of  the  principal 
planes ;  indeed,  it  may  be  said  tliat  they  are  probably  more  often 
vertical  than  otherwise. 

.53.  The  use  of  such  a  plane  is  j^it^torially  represented  in  Fig. 
60,  in  which  OR  is,  let  us  say,  an  oblicpiely  placed  wire,  suj^ported 


Fig.  60 


by  two  vertical  ones  fixed  in  the  horizontal  plane.  There  are 
shown  the  projections  of  these  not  only  upon  H  and  T,  but  uj^on  a 
supplementary  plane  S,  in  this  case  parallel  to  the  horizontal  jiro- 
jecting  plane  Or.  The  three  lines  are  therefore  projected  upon  S 
in  their  true  lengths  and  relative  positions,  while  upon  II  and  V  tliey 
are  not ;  and  it  is  for  tlie  purpose  of  thus  convejdng  directly  in- 
formation which  the  other  views  do  not  give  explicitly,  that  such 
supplementary  projections  are  chiefly  employed. 

It  is  obvious  that  in  viewing  S  perpendicularly,  as  indicated  by 
the  arrow,  the  axis  of  vision  is  parallel  to  H,  which,  being  thus  seen 
edgewise,  will  in  projection  be  represented  by  a  line  bearing  the 
same  relation  to  this  new  drawing  that  the  original  ground  line 
bears  to  the  vertical  projection. 

54.  This  is  shown  in  Fig.  61,  where  <?r,  o'r'  are  the  projections 
upon  H  and  V  of  a  limited  oblique  line,  like  OR  in  Fig,  60.  Be- 
low is  drawn  a  supplementary  view,  looking  perpendicularly 
against  the  horizontal  projecting  plane,  as  shown  by  the  arrow  a. 

The  horizontal  plane  is  represented  by  the  line  II' H'  perpen- 
dicular to  the  arrow,  and  the  points  o  andr,  being  projected  per- 
pendicularly toward  this  line,  appear  in  the  new  view  at  distances 


DESCRIPTIVE    GEOMETRY.  31 

above  it  equal  to  those  of  o'  and  r'  above  AB.  To  use  a  geo- 
grapliical  illustration,  if  the  projection  upon  V  be  regarded  as  a 
view  from  the  south,  the  observer  looking  due  north,  this  supple- 
mentary jDrojection  is  a  view  of  the  same  object  from  the  south- 
west, the  observer  looking  northeast ;  bj  this  change  of  position  on 
the  part  of  the  spectator  tlie  altitudes  of  the  various  parts  of  the 
object  are  clearly  neither  increased  nor  diminished.  Also,  the  line 
OR  is  now  seen  in  its  true  length. 

55.  But  again,  the  object  still  retaining  its  original  position, 
the  eye  may  be  sup|)Osed  to  be  above  and  at  the  same  time  either 
to  the  right  or  to  the  left  of  it,  and  to  be  directed,  not  vertically,  but 
obliquely  dowuAvard,  yet  still  in  a  line  parallel  to  V.  In  this  case 
the  vertical  plane  will  be  seen  edgewise,  but  the  vertical  projecting 
lines  will  remain  unchanged  in  length,  so  that  all  the  points  of  the 
object  will  appear  just  as  far  from  that  plane  as  in  the  original 
horizontal  projection. 

In  illustration,  a  supplementary  view  is  given  in  the  upper  part 
of  Fig.  61,  looking  in  the  direction  shown  by  the  arrow  h.  The 
line  y  F',  perpendicular  to  the  arrow,  represents  the  vertical 
plane  toward  which  the  points  o' ^  r'  are  projected,  their  distances 
from  y  y  being  equal  to  those  of  the  horizontal  projections  6>,  r, 
from  AB. 

56.  Such  supplementary  projections,  like  the  profiles,  should 
always  be  constructed  as  detached  and  independent  view^s;  their 
precise  location  is  of  course  arbitrary,  but  should  always  be  such  as 
to  j^revent  the  possibility  of  confounding  the  lines  with  those  of  the 
other  views. 


d2  DESCRIPTIVE    GEOMETRY. 


CHAPTER  II. 

ELEMENTARY    PROBLEMS    RELATING    TO    THE    POINT,    RIGHT    LINE,    AND 

PLANE. 

57.  It  is  necessary  to  make  a  clear  distinction  between  tlie 
solution  of  a  problem  and  tlie  representation  of  that  solution. 

Tlie  solution  is  effected  by  abstract  reasoning :  one  link  after 
another  being  added  to  a  cliain  of  logical  ai-guments  until  a  detinite 
conclusion  is  reached  which  demonstrates  that  the  object  sought 
can  be  accomplished  in  a  certain  way.  This  is  a  purely  mental 
process ;  clear  conceptions  can  be  formed  in  the  dark,  or  by  a  blind 
man,  of  the  magnitudes  involved,  of  their  relations  to  each  other, 
of  the  various  steps  to  be  taken  and  their  results — in  short,  of  the 
complete  solution  of  any  problem ;  which  is  wholly  independent  of 
its  representation  and  of  any  graphic  operation  whatever. 

The  processes  of  descriptive  geometry,  on  the  other  hand,  are 
purely  graphic.  And  it  is  the  province  of  this  science  to  explain 
the  methods,  not  of  solving  j^roblems,  but  of  exactly  representing 
the  data,  steps,  and  results  of  solutions  already  effected  by  mathe- 
matical reasoning.  This  distinction  is  natural  and  inevitable,  be- 
cause before  a  thing  can  be  represented  it  must  be  known  what 
that  thing  is. 

58.  Analysis  and  Construction.  A  problem  being  enunciated, 
then,  its  treatment  will  consist  of  two  distinct  parts.  First^  a  clear 
statement  of  the  principles  and  reasoning  employed  in  the  solution 
and  applied  to  the  magnitudes  in  space ;  this  is  the  analysis.  Second y 
an  explanation,  in  due  order,  of  the  lines  employed  in  representing^ 
on  paper,  the  problem  and  its  solution ;  this  is  called  the  construction 
of  the  problem. 

59.  Method  of  Study.  The  same  processes  may  in  general  be 
applied  to  magnitudes  under  widely  varying  conditions;   and  in  the 


DESCKIPTIYE    GEOMETRY. 


33 


nature  of  tilings  but  a  limited  number  of  eases,  and  often  only  one, 
can  be  worked  out  in  illustration.  Consequently  great  care  should 
be  taken  to  avoid  associating  the  solution  of  any  problem  with  tlie 
aj^pearance  of  the  figure,  because  the  assumption  of  different  data 
may  result  in  the  production  of  a  totally  dissimilar  diagram.  This 
is  of  especial  importance  in  regard  to  these  elementary  problems, 
because  they  are  subsequently  to  be  used  as  mere  steps  in  the  solu- 
tion of  more  complex  ones,  and  the  conditions  thus  fixed  may  be 
quite  unlike  those  previously  met  with.  The  best  course  therefore 
is  to  di823ense,  as  soon  and  as  far  as  possible,  with  all  reference  to 
the  illustrations ;  first  mastering  the  analysis  and  fixing  in  mind  the 
successive  stej)s,  and  then  making  an  original  construction  by  apply- 
ing them  to  assumed  data. 

60.  Problem  1.  To  find  the  true  distance  between  two  imlnt^ 
given  hy  their  jyrojections. 

Analysis.  The  required  distance  is  the  length  of  the  right  line 
joining  the  two  points.  If  either  projecting  plane  of  this  line  be 
revolved  about  its  trace  into  the  corresponding  plane  of  projection, 
the  line  will  be  seen  in  its  true  length. 

Construction.  This  is  represented  pictorially  in  Fig.  62,  and 
Fig.  63  shows  it  in  projection,  M  and  N  being  the  given  points. 
Revolving  the  horizontal  projecting  plane  of  MN  about  nm  into  H, 


Fig.  62 


liie  point  M  goes  to  m" ;  mm"  being  perpendicular  to  mn  and  equal 
to  tn'x^  the  horizontal  projecting  line.  Similarly,  N  goes  to  n" ^  the 
distance  nn"  being  equal  to  n'y\  and  m"n"  is  the  distance  required. 
The  required  distance  may  also  be  ascertained  by  means  of  a 
supplementary  projection,  as  shown  in  Fig.  61,  or  by  means  of  an 


34 


DESCRIPTIVE   GEOMETRY. 


independent  construction,  apart  from  tlie  drawing,  as  explained  in 
(15). 

JS^.B.  In  this  case  the  distances  tnm"^  nn'\  are  set  off  in  op23osite- 
directions,  because  the  points  M  and  N  are  on  opposite  sides  of  the 
axis.  Had  tliej  been  on  the  same  side,  as  M  and  P  are,  these  dis- 
tances would  have  been  set  off  in  the  same  direction.  In  either 
case,  the  line  joining  the  two  points,  if  not  parallel  to  the  plane 
into  which  it  is  revolved,  will  pierce  it  if  j)rolonged,  as  at  o^  o'  in 
Fig.  63.  This  point  of  penetration,  being  in  the  axis,  remains 
fixed,  and  the  given  line  must  pass  through  it  in  its  revolved  as 
well  as  in  its  original  position;   thus  7irh"n"  passes  through  o. 

61.  If  it  is  required  to  set  off  from  Jf  a  distance  along  J/iV^ 
equal  to  a  given  line  cd^  first  revolve  the  line  MN  into  H  as  above, 
and  then  lay  off  m!'p"  equal  to  cd.  In  the  counter-revolution,  j)" 
goes,  in  a  direction  perpendicular  to  mn^  to  the  position^,  ^vhich  is 
vertically  projected  at  p'  on  m'n\  This  operation  is  identical  with 
that  represented  in  Figs.  56  and  57. 

62.  Problem  2.  To  jpass  ajplane  through  three  given  points 
tiot  in  the  same  right  line. 

Analysis.  Through  either  tw^o  of  tlie  points  draw  a  right  line. 
Through  the  third  point  draw  another  right  line,  either  parallel  to 
the  first  or  intersecting  it  at  any  point.  These  two  Ihies  determine 
the  plane,  and  their  traces  wiU  be  points  in  the  corresponding  traces 
of  tlie  plane. 

Construction.     In  Figs.  64  and  'oOy  C\  J),  and  J^  are  the  given 


Fig.  64  "^  Fig.  65 

points.     Draw  CD  and  produce  it  to  pierce  II  at  6>,  and  V  at  Jf ; 

then  <9  is  a  point  in  the  horizontal  and  m'  a  point  in  the  vertical 
trace  of  the  required  plane.     Join  the  third  point  E  with  any  point 


DESCRIPTIVE   GEOMETRY.  35 

G  of  OM^  and  produce  EG  to  pierce  H  and  V  at  7?  and  N\  then  r 
is  another  point  in  tlie  horizontal  and  n!  another  point  in  the  verti- 
cal trace.  Therefore  n'm' ^  or  are  tlie  required  traces,  which, 
when  produced,  must  meet  in  the  ground  line,  unless  tliej  are 
parallel. 

Note.  — The  direction  of  the  second  line  EG  should  be  so  chosen 
that  the  distance  between  o  and  r^  and  also  that  between  /?/  and  m\ 
shall  be  as  great  as  possible. 

63.  The  problems  of  drawing  a  plane  through  one  right  line 
and  parallel  to  another,  and  of  drawing  a  plane  through  a  given 
point  parallel  to  two  given  right  lines,  are  scarcely  more  than  varia- 
tions of  the  preceding  one ;  for,  in  the  first  case,  we  have  already 
one  line  of  the  required  plane  and  know  the  direction  of  another, 
w^hich  may  be  drawn  through  any  given  point  of  the  given  line ; 
and  in  the  second  case,  we  know  the  directions  of  two  lines  of  the 
required  plane  and  have  merely  to  draw  them  through  a  given 
point  and  find  their  traces.  If  either  line  be  parallel  to  AB,  the 
plane  itself  and  both  its  traces  will  be  parallel  to  the  ground  line. 
In  this  case  a  profile  should  be  drawn,  in  addition  to  the  projections 
on  II  and  V. 

64.  Problem  3.  To  draw  through  a  given  jpoint  a  plane  per- 
'pendieular  to  a  given  right  line. 

Analysis.  Tlie  directions  of  the  traces  are  known,  being  respec- 
tively perpendicular  to  the  projections  of  the  line  (43).  Draw 
through  the  given  point  a  line  parallel  to  either  trace ;  this  wiU 
be  a  line  of  the  plane,  and  will  pierce  the  other  plane  of  projection 
in  a  point  of  the  required  trace  upon  that  plane.  This  trace,  being 
perpendicular  to  the  corresponding  projection  of  the  line,  may  now 
be  drawn ;  it  will  cut  the  ground  line  in  a  point  of  the  remainiDg 
trace,  of  which  the  direction  is  also  known. 

Construction.  Let  P,  Figs.  QQ  and  67,  be  the  given  point  and 
MN  the  given  line.  Draw  through  P  a  line  parallel  to  the  hori- 
zontal trace  of  the  required  plane;  its  horizontal  projection  isj[?o, 
perpendicular  to  mn^  and  its  vertical  projection  \s,  j[>'o\  parallel  to 
AB.  This  is  a  line  of  the  plane,  and  its  vertical  trace  0  is  a  point 
in  the  vertical  trace  of  the  plane.     Therefore  t'o'T^  perpendicular 


36 


DESCRIPTIVE    GEOMETRY. 


to  m'n',^  is  that  vertical  trace,  Avliicli  cuts  AB  at  T\  and  Tt^  per* 
pendicular  to  inn^  is  the  horizontal  trace. 

N.  B.  If  the  projections  of  the  given  hne  coincide,  as  in  Fig.  29, 
the  traces  of  the  plane  will  also  coincide,  like  those  of  the  plane 
tTt'  in  Fig.  40,  If  the  given  line  lie  in  a  profile  plane,  the  re- 
quired plane  will  be  ]3arallel  to  the  gronnd  line.  Thus  in  Fig.  ^'^^ 
P  is  the  given  point,  MN  the  given  line ;  these  are  seen  in  their 
true  relations  to  H  and  V  in  the  prohle,   Fig.  69,  where  a  per- 


n 

\ 

*P 

m 

Fig.  68 

Fig.  69 

pendicular  to  MN  through  P  represents  the  required  plane,  cutting 
V  in  ^'  and  H  in  ^;  these  points  are  the  profile  projections  of  the 
traces  tt^  t't^  in  Fig.  ^'^. 

65.  The  reasoning  in  the  analysis  of  this  problem  is  precisely 
the  same  as  that  used  (41)  in  reference  to  the  drawing  of  a  plane 
through  a  given  point  and  parallel  to  a  given  plane,  the  construc- 
tion of  which  was  shown  in  Fig.  50.  The  gist  of  the  argument  is 
Bimply  this,  that  when  the  directions  of  the  ti-aces  are  known,  the 
location  of  a  single  point  in  either  trace  determines  the  plane,  ex- 
cept when  it  is  parallel  to  the  ground  line  \  in  that  case  a  point  in 
each  trace  must  be  found. 


DESCRIPTIVE    GEOMETRY. 


37 


66.  Problem  4.    To  find  tJie  intersection  of  two  flanes. 

Analysis.  The  intersection  of  the  vertical  traces  will  be  one 
point,  and  the  intersection  of  the  horizontal  traces  will  be  anotiier, 
in  the  required  line,  which  is  determined  by  those  two  points.  If 
both  planes  are  perpendicular  to  either  of  the  principal  planes,  their 
traces  on  the  other  jDlane  will  be  parallel  to  each  other  and  to  the 
required  line,  which  Avill  pass  through  the  intersection  of  the  other 
two  traces.  If  both  planes  are  parallel  to  the  ground  line,  the  re- 
quired line  will  be  so  likewise ;  it  is  determined  by  tlie  intersection 
of  the  profile  traces  of  the  given  planes. 

Construction.     In  Figs.  70  and  71,  sSs\  tTt'  are  the  two  planes. 


Fig.  72 


Fig.  74 


Fig.  73 


The  horizontal  traces  intersect  at  c,  whose  vertical  projection  is  <f 
in  AB,  and  the  vertical  traces  intersect  at  d' ^  whose  horizontal  pro- 
jection is  d  in  AB;  therefore  cd  is  the  horizontal  and  c'd'  the  ver- 
tical projection  of  the  required  intersection. 

AYhen  both  planes  are  vertical,  their  intersection  is  vertical  and 
passes  through  the  intersection  of  the  horizontal  traces,  as  shown  in 


Fig. 


<2,  at  the  left;  when  the  planes  are  perpendicular  to  V,  as 


38 


DESCRIPTIVE    GEOMETRY. 


shown  in  the  same  figure  at  the  right,  tlieir  intersection  is-  also  per- 
pendicular to  y  and  passes  through  the  intersection  of  the  vertical 
traces. 

When  both  planes,  and  consequently  their  intersection,  are 
jDarallel  to  AB,  the  detached  profile.  Fig.  73,  shows  the  condition  of 
things  with  perfect  distinctness ;  but  the  projection  on  the  princi- 
pal planes.  Fig.  74,  is  bj  itself  simply  useless  as  a  means  of  impart- 
ing information. 

67.  Some  Special  Cases  of  the  Above  Prchlein. — In  Fig. 
75  the  horizontal  traces  do  not  intersect  within  the  limits  of  the  draw- 
ing ;  but  one  point,  i>,  of  the  required  Kne  is  determined  by  the 
intersection  of  the  vertical  traces.  In  order  to  ascertahi  its  direc- 
tion, draw  an  auxiliary  plane  ILL  \  parallel  to  tTt'  and  cutting  sSs' 


Fig.  77 


Fig.  78 


Fig.  79 


in  the  line  JfiT,  found  as  in  Fig.  71.  Tliis  intersection  is  parallel 
to  the  one  sought,  of  which,  therefore,  the  vertical  projection  is  c'd\ 
parallel  to  r)i'n\  and  cd,  parallel  to  mn^  is  its  horizontal  projection. 
In  Fig.  76  the  intersections  of  the  vertical  traces  and  of  the 
horizontal  traces  are  both  inaccessible.  Draw  an  auxiliary  horizon- 
tal plane,  of  which  Jc'h'  is  the  trace.      This  cuts  the  plane  tTt'  in  a 


DESCRIPTIVE    GEOMETRY.  3.9 

line  of  which  one  point  is  r'  on  Y,  horizontally  projected  at  r  on 
AB ;  this  line,  being  horizontal,  is  parallel  to  the  horizontal  trace 
and  its  horizontal  projection  is  therefore  drawn  through  r  and  par- 
allel to  Tt.  The  auxiliary  plane  also  cuts  sSs'  in  a  line  whose  hori- 
zontal projection  is  drawn  through  o,  parallel  to  Ss.  These  two 
lines,  one  in  eacli  given  plane,  cut  each  other  in  a  point  of  which 
the  horizontal  projection  is  <?,  and  the  vertical  projection  is  c'  on 
k'k' .  Thus  one  point  in  the  required  line  is  determined,  and  its 
direction  is  ascertained  as  in  Fig.  75. 

In  Fig.  77  the  traces  of  the  plane  tTt  coincide  as  in  Fig.  40, 
and  sSs'  is  a  profile  plane  cutting  the  ground  line  at  the  same  point ; 
in  Fig.  78  both  planes  are  oblique,  with  coincident  traces.  Draw- 
ing in  each  case  an  auxiliary  plane  parallel  to  sSs^  as  in  Fig.  75, 
the  line  MN  cut  from  the  j)lane  tTt'  is  parallel  to  the  required 
intersection.  And  in  each  case  this  line  pierces  H  behind  Y,  and 
Y  above  H,  in  points  equally  distant  from  AB ;  it  therefore  crosses 
the  second  angle,  as  shown  in  the  profile.  Fig.  79,  and  is  equally 
inclined  to  H  and  Y.  The  required  line  CD^  being  parallel  to  MN 
and  intersecting  AB,  therefore  lies  in  a  profile  plane  and  bisects  the 
first  and  third  angles ;  its  projections  in  Fig.  77  coincide  in  S8\  and 
in  Fig.  78  they  coincide  in  a  line  through  T,  perpendicular  to  AB. 

68.  Problem  5.  To  find  the  ^oint  in  which  a  gwen  right  line 
'pierces  a  given  plane. 

Analysis.  Pass  any  plane  through  the  given  line  and  find  its 
intersection  with  the  given  plane.  This  line  will  cut  the  given  line 
in  tlie  required  point. 

Construction.  In  Figs.  80  and  81,  MN  \s>  the  given  line,  tTt' 
the  given  plane.  Since  any  plane  containing  MN  will  serve  our 
purpose,  we  use  for  convenience  one  of  its  projecting  planes;  in 
this  case  the  horizontal.  Its  horizontal  trace  coincides  with  the 
horizontal  projection  of  the  line,  and  its  vertical  trace  is  perpen- 
dicular to  AB ;  it  intersects  tTt'  in  the  line  (7i>,  whose  vertical  pro- 
jection c'd'  cuts  the  vertical  projection  m'n'  in  <?',  the  vertical  pro- 
jection of  the  required  point ;  the  horizontal  projection  is  6>,  on  cd. 

N.  B.  Had  the  intersection  at  o'  been  very  acute,  the  deter- 
mination would  have  been  less  reliable ;  "and  a  better  result  might 
have  been  obtained  by  using  the  vertical  projecting  plane,  thus  de* 


40 


DESCRIPTIVE    GEOMETRY. 


terminiiig  first  the  horizontal  projection  o  of  the  required  point.  It 
is  not  certain  that  this  would  happen,  since  if  the  line  were  but 
slightly  inclined  to  the  plane,  both  these  intersections  would  be 
acute ;   in  which  case  both  determinations  should  be  made,  and  if 


JFiG.Sl 


Fig.  80 


they  do  not  agree,  a  mean  between  them  may  be  taken  as  the  cor* 
rect  result. 

69.  The  preceding  construction  involves  the  use  of  the  traces 
of  the  given  plane :  but  if  two  lines  of  a  plane  are  given,  it  is  not 
necessary  to  dnd  the  traces  in  order  to  determine  the  j)oint  in  which 
it  is  pierced  by  a  third  line.      Thus  in  Fig.  82,  let  it  be  required  to 


Fig.  82 


Fig.  83 


find  the  point  in  which  the  line  MN  pierces  the  plane  determined 
by  the  two  intersecting  lines  Kl^  Gl.  Using  again  for  conven- 
ience the  horizontal  projecting  plane  mdd'  of  the  given  line,  it  cuts 


DESCiUPTlVE    GEOMETRY. 


41 


KI  in  E^  and  GI  in  F\  EF  therefore  lies  in  both  planes,  and  tlie 
point  0  in  which  it  intersects  MN^  is  the  required  point.  The 
traces  of  the  given  j)lane  are  shown  in  this  pictorial  representation, 
for  tlie  purpose  of  calling  attention  to  the  fact  that  EF\^  merely  a 
portion  of  the  same  line  of  intersection  GD^  which  was  determined 
in  Fig.  80  by  means  of  the  traces  of  the  two  planes. 

The  construction  in  projection  is  given  in  Fig.  83,  where  GR^ 
KL^  intersecting  at  /,  determine  a  plane,  and  it  is  required  to  find 
the  point  in  which  this  plane  is  pierced  by  the  line  MN.  The 
horizontal  trace  mn^  of  the  horizontal  projecting  plane,  must  con- 
tain the  horizontal  projections  of  all  lines  and  points  that  lie  in  it, 
because  the  plane  is  vertical.  And  mn  cuts  gr  at/*,  which  is  the 
horizontal  projection  of  a  point  on  GR^  whose  vertical  projection 
is/"  on  g'r' .  Similar Iv,  the  line  KL  is  seen  to  pierce  the  project- 
ing plane  in  a  point  whose  horizontal  projection  is  e^  and  whose 
vertical  projection  is  e'  on  h'V .  Consequently  e'f  is  the  vertical 
projection  of  the  portion  of  the  line  of  intersection  thus  determined ; 
it  cuts  m'n  in  o\  the  vertical  projection  of  the  required  point,  and 
the  horizontal  projection  is  o  on  mn, 

70.  Some  Sjyecial  Cases  of  the  Above  Problem.  In  Fig.  84, 
the  given  plane  is   parallel    to   AB,    and  the  projections   of   the 


Fig.  86, 


V 

i 

o' 

N 
\ 

H 

/y  0 

Fig.  87 

•  V 

given  line  MN  coincide.     The  horizontal  trace  of  the  horizontal 
projecting  plane  cuts  tt  at  <?,  vertically  projected  at  g'  in  AB ;  its 


42  DESCRIPTIVE    GEOMETRY. 

vertical  trace  cuts  t't'  at  d\  of  whicli  d  is  the  horizontal  projection; 
and  c'd'  intersects  Tn'n'  in  o\  the  vertical  projection,  which  coin- 
cides with  6>,  the  horizontal  projection  of  the  required  point. 

In  Fig.  85,  the  traces  of  tTt'  coincide,  and  MN  is  parallel  to 
AB.  The  horizontal  projecting  plane  cuts  tTt'  in  a  line  parallel  to 
y  aivd  therefore  to  Tt'  \  one  point  of  this  line  is  determined  by  the 
intersection  of  the  horizontal  traces  at  6',  vertically  projected  at  c' 
in  AB,  and  its  vertical  projection  c'o'  cuts  m'n'  at  o\  of  which  o  on 
7nn  is  the  horizontal  projection. 

In  Fig.  '^^^  the  two  projections  of  J/LZTare  nearly  perpendicular 
to  AB.  In  such  cases  the  direct  determinations  by  the  method* 
before  explained  are  apt  to  be  very  unreliable  on  account  of  the 
acuteness  of  the  intersections :  and  the  profile  may  be  used  to  great 
advantage  in  the  manner  here  i]lustrated.  In  this  instance  the 
vertical  projecting  plane  of  the  given  line  has  been  used ;  its  verti- 
cal trace  cuts  Tt'  at  x' ^  and  the  ground  line  at  L ;  its  horizontal 
trace  is  perpendicular  to  AB  and  cuts  Tt  at  ?/.  Tlie  line  of  inter- 
section will  therefore  pass  through  x'  on  V  and  y  on  H ;  but  its 
projection  on  the  latter  is  not  drawn.  In  drawing  the  profile,  Fig. 
87,  x'  is  projected  horizontally  across  from  Fig.  86,  and  the  distance 
of  ?/  from  FF  is  equal  toZy  in  the  horizontal  projection;  then  x'y 
represents  the  line  of  intersection.  In  this  particular  case  MN 
pierces  V  ati\'^,  therefore  n'  is  projected  directly  across  to  W\ 
the  altitude  of  M  is  the  same  in  both  views,  and  so  is  its  distance 
from  V ;  and  MN  in  the  profile  intersects  x'y  in  0^  which  being 
projected  back  to  in'n!  in  Fig.  86,  determines  o'  the  vertical  projec- 
tion of.  the  required  point.  Tlie  distance  of  0  from  T  is  seen  in 
the  profile ;  and  drawing  in  Fig.  ^^^  a  parallel  to  AB  at  that  dis- 
tance from  it,  the  horizontal  j)rojecti()n  o  is  determined  much  more 
accurately  than  it  could  be  by  drawing  through  o'  a  perpendicular 
to  AB. 

Should  the  given  line  lie  in  a  plane  perpendicular  to  AB,  tlic 
construction  of  a  profile  is  of  course  a  necessity. 

71.  Problem  6.  To  find  the  distance  of  a  given  ^oint  from 
a  given  jplane. 

Analysis.     ].   Draw  through  the  point  a  perpendicular  to  tlie 


DESCRIPTIVE    GEOMETRY. 


43 


plane.      2.   Find  tlie  point  in  wliicli  it  pierces  tlie  plane.      8.   Find 
the  distance  between  tins  point  and  the  given  point. 

Construction.     In   Fig.  88,  let  P  be  the  given  point,  tTt'  the 
given  plane.     Draw  through  P  a  perpendicular  to  tTf  as  in  Fig. 


4 
e 

„ .^ — 7*" 

c/'     >'       t/ 

/ 

d 

\JrKf 

V         \p       FiG.89 

Fig.  88 


52.  Find  the  point  0  in  which  it  pierces  the  plane  as  in  Fig.  81. 
Find  the  true  length  oi  PO  as  in  Fig.  63. 

In  Fig.  89,  the  direction  of  the  vertical  trace  Tf  is  the  same  as 
in  Fig.  88,  but  that  of  the  horizontal  trace  2Y  is  different.  In  con- 
sequence of  this  change,  the  vertical  trace  of  the  horizontal  pro- 
jecting plane  cuts  that  of  the  given  plane  at  a  point  d'  below  AB  in- 
stead of  above  it  as  before,  and  d^c'  must  be  produced  to  detennine 
g\  PO  is  here  revolved  into  V  instead  of  H;  it  pierces  Y  at  a 
point  of  which  the  horizontal  projection  is  ^,  and  the  vertical  pro- 
jection is  e'  on  p'o^  produced ;  and  since  e'  is  on  the  axis,  it  re- 
mains fixed,  and  the  prolongation  oi  p"o"  passes  through  it. 

When  the  given  plane  is  parallel  to  AB,  the  required  distance  is 
found  directly  by  constructing  a  profile. 

72.  Problem  7.  To  jproject  a  given  right  line  tipon  a  given 
plane. 

Analysis.  Through  any  point  of  the  given  line,  draw  a  per- 
pendicular to  the  given  plane  :  these  two  lines  determine  a  second 
plane,  perpendicular  to  the  first.  The  intersection  of  these  two 
planes  is  the  required  projection. 

Construction.  In  Fig,  90,  let  KG  be  the  given  line,  tt  the  hori- 
zontal trace,  and  t't'  the  vertical  tra(;e  of  the  given  plane.  From  any 
point  P  on  KG^  draw  a  perpendicular  to  the  plane ;   the  traces  of 


44 


DESCJtiPTiVE  geomp:try. 


this  perpendicular  are  X  and  Z,  and  the  traces  of  the  given  Hne  are 
TJ  and  Y.  Therefore  112  is  the  horizontal  and  x'y'  is  the  vertical 
trace  of  tlie  plane  sSs  ^  determined  by  the  given  line  KG  and  the 
projec  ing  perpendicular  PX.  This  plane  cuts  the  given  plane  in 
the  line  CD^  which  is  the  required  projection. 


73.  This  intersection  CD  evidently  must  contain  the  point  TT, 
in  which  the  given  line  pierces  the  plane  tTt' ^  and  also  the  point  (9, 
which  is  the  foot  of  the  perpendicular  let  fall  upon  the  plane  from 
the  point  P ;  and  ON  is  the  projection  of  the  hypothenuse  PN  of 
the  right-angled  triangle  PON. 

The  points  0  and  N  might  have  been  found  as  in  Problem  5, 
without  determining  the  traces  of  sSs  .  And  if  the  projection  of  a 
definite  portion  of  the  line,  as  for  instance  PG  in  the  figure,  is 
required,  two  perpendiculars,  as  PO^  GA,  may  be  drav/n,  and 
the  points  of  penetration,  0  and  A^  found  in  the  same  way; 
indeed,  this  may  be  necessary,  if  the  line  is  but  slightly  inclined  to 
the  plane.  The  three  methods  are  identical  in  j^i'inciple,  and  the 
selection  must  depend  upon  considerations  of  convenience,  deter- 
mined by  the  given  conditions  in  any  particular  case. 

If  the  giTen  line  be  parallel  to  the  given  plane,  its  projection  on 


DESCRIPTIVE    GEOMETRY. 


45 


that  plane  will  be  parallel  to  the  line  itself  (14).  Therefore  the 
required  projections  will  be  parallel  to  those  of  the  given  line,  and 
the  determination  of  one  point  in  each  is  sufficient. 

74.  Some  Special  Cases  of  the  Above  Problem.  In  Fig.  91, 
tTt'  is  the  given  plane;  and  the  given  line  KG  is  parallel  to 
AB.  Draw  througli  any  point  P  on  GK 2,  perpendicular  to  tTt'  \ 
it  pierces  the  horizontal  plane  in  Z  and  the  vertical  plane  in 
X.  The  plane  of  these  two  lines  is  parallel  to  AB,  therefore  its 
traces  are  sz  and  s'x  ^  also  parallel  to  AB ;  and  it  cuts  tTt'  in  the 
line  CD^  the  required  projection. 

In  Fig.  92,  tTt'  is  the  given  plane ;  it  is  required  to  project  the 
ground  line  on  it.  From  any  point  P  on  AB  draw  Pr  perpen- 
dicular to  Tt  and  Pr'  perpendicular  to  Tt' ;  these  are  the  pro- 
jections of  a  line  perpendicular  to  the  plane.  The  vertical  pro- 
jecting plane  of  this  line  cuts  tTt'  in  the  line  (72>,  which  intersects 
PR  in  (9,  the  projection  of  the  point  P  upon  the  given  plane. 
That  plane  cuts  AB  in  the  point  T\  consequently  To  is  the  hori- 
zontal and  To'  is  the  vertical  projection  of  the  required  line. 

In  Fig.  93,  the  given  line  is  inclined  to  both  planes,  piercing  H 
in  the  point  ^and.V  in  the  point  Y.  The  given  plane  being  par- 
allel to  the  ground  line,  the  perpendicular  to  it  from  the  point  P 


Fig.  94 


will  lie  in  a  plane  perpendicular  to  AB ;  its  traces  are  readily  deter- 
jnined  by  drawing  the  profile,  Fig.  94  ^  then  setting  off  z  and  x'  in 
Fig.  93  at  the  distances  from  AB  thus  found,  we  have,  as  before, 


46 


DESCRIPTIVE    GEOMETKY. 


uz  for  the  horizontal  and  x'y'  for  the  vertical  trace  of  the  plane 
sSs\  which  cuts  tTt'  in  the  Hue  CD^  the  required  projection. 

75.  Problem  8.  To  find  the  distance  of  a  given  point  from  a 
given  line. 

First  Method.  Analysis.  1.  Through  the  given  point  pass  a 
plane  perpendicular  to  the  given  line.  2.  Find  the  point  in  which 
the  given  line  pierces  this  plane.  3.  Find  the  distance  between 
this  point  and  the  given  point. 

Construction.  In  Fig  95,  let  P  be  the  given  point,  KG  the 
given  line.     Draw  through  P  a  plane  tTt'  perpendicular  to  KG^  as 


Pig.  96 

in  Fig.  67.     Find  the  point  0,  in  which  KG  pierces  tTt\  as  ip 
Fig.  81.     Find  the  length  of  PO  as  in  Fig.  63. 

Special  Case.  In  Fig.  96,  the  two  projections  of  P  coincide, 
as  do  those  of  KG ;  consequently  the  traces  of  the  perpendicular 
plane  tTt'  also  coincide.  The  horizontal  projecting  plane  of  KG 
cuts  tTt'  in  the  line  CD^  which  intersects  KG  in  0.  The  line  KG 
lies  in  a  plane  bisecting  the  second  and  fourth  angles,  and  cuts  AB 
at  iV";  the  line  OP  also  lies  in  that  bisecting  plane,  and  since  it  is 
at  the  same  time  a  line  of  the  plane  tTt\  it  will  when  produced  cut 
AB  at  T.  Therefore  NT  is  the  hypothenuse  of  the  right-angled 
triangle  TOK\  and  when  this  triangle  is  revolved  about  AB  into 
either  V  or  H,  0  will  fall  at  o"  on  the  circumference  of  a  semicircle 
of  which  NT \&  the  diameter,  and  /-*  falls  at/)"  on  o"T. 


DESCRIPTIVE    GEOMETRY. 


47 


76.  Second  Method.  Analysis.  Througli  the  given  point  draw 
a  line  either  parallel  to  or  intersecting  the  given  line.  Revolve  the 
plane  of  these  two  lines  about  one  of  its  traces  into  tlie  corresponding 
plane  of  projection ;  the  line  and  point  will  then  be  seen  in  their 
true  relative  positions.  A  perpendicular  from  this  revolved  position 
of  the  point  to  that  of  the  line  will  be  the  required  distance. 

Construction.  In  Fig.  97,  P  is  the  given  point,  KG  the  given 
line.  Draw  through  P  a  parallel  to  KG ;  it  pierces  Y  in  iV,  and 
KG  pierces  it  in  M.     Eevolve  this  plane  about  its  vertical  trace 


Fig.  98 


Tim'  into  Y\  P  goes  to p" ^  and  as  n'  remains  fixed,  being  in  the 
axis,  jp"n'  is  the  revolved  position  of  the  second  line.  And  since 
the  two  lines  are  parallel,  a  line  through  m',  parallel  to  p"n\  is  the 
revolved  position  of  the  given  line,  and  Jp"o" ^  perpendicular  to  it, 
is  the  actual  distance  required. 

In  the  counter-revolution,  jp"  returns  to  jp\  and  o"  goes,  in  a 
direction  perpendicular  to  the  axis  mn\  to  the  position  o'  on  Tn'h' . 
This  is  the  vertical,  and  o  on  hm  is  the  horizontal,  projection  of  <?, 
the  foot  of  the  perpendicular  drawn  from  the  given  point  P  to  the 
given  line. 

Special  Case,  In  Fig.  98,  the  projections  of  the  given  line 
KG  coincide,  and  the  given  point  P  is  in  the  vertical  plane.  There 
is  a  point  on  KG  whose  projections,  c  and  c\  coincide  with  p' .  The 
vertical  projecting  line  of  this  point  therefore  passes  through  the 
given  point,  and,  with  the  given  line,  determines  a  plane  of  which 
c' N  \'&  the  vertical  trace.      Revolving  this  plane  about  this  trace 


48 


DESCRIPTIVE    GEOMETRY. 


into  V,  C  goes  to  g'\  c" N  is  the  revolved  position  of  the  given  hne, 
and  J?'  is  stationary.  Therefore  j^'o"  perpendicular  to  c" N  is  the 
required  distance.  In  the  counter-revolution  g"  returns  to  c\  o" 
goes  to  o'  on  Tt'g'^  and  jpo^  jp'o\  are  the  projections  of  the  perpen- 
dicular. 

77.  Problem  9.  To  find  the  angle  hetween  two  lines ^  and  ta 
divide  it. 

Analysis.  If  the  plane  of  the  two  lines  be  revolved  about  one 
of  its  traces,  or  a  line  parallel  thereto,  until  it  coincides  with  or  iz 
parallel  to  the  corresponding  plane  of  projection,  the  angle  will  be 
seen  in  its  tnie  size  and  may  be  subdivided. 

Otherwise :  If  a  supplementary  projection  be  made  upon  a. 
plane  parallel  to  that  determined  by  the  given  lines,  the  angle  will 
appear  in  its  true  size. 

Construction.  In  Fig.  99,  the  two  lines  which  intersect  in 
0  pierce  H  in  the  points  P  and  N.     Eevolving  them  about  the 

horizontal  trace  7i/p  of  their  plane 
into  H,  their  intersection  0  goes- 
to  o'\  the  distance  xo"  being  equal 
to  the  hypotlienuse  of  a  triangle  of 
which  ox  is  the  base  and  o'y  the 
altitude.  Tlius  the  angle  is  seen 
in  its  true  size  as  no"2y ;  if  it  be 
now  required  to  divide  it,  for  in- 
stance into  two  equal  parts,  the 
bisecting  line  cuts  njp  at  r^  which 
in  the  counter-revolution  will  re- 
main stationary  :  therefore  07\  o'r\ 
are  the  projections  of  the  bisector.. 

78.  In  Fig.  100,  the  horizontal  trace  of  OP  is  inaccessible. 
Draw  therefore  an  auxiliary  horizontal  plane  AA,  cutting  the  given 
lines  in  C  and  .Z>,  and  revolve  the  lines  about  cd  until  they  are 
parallel  to  H ;  in  this  revolution  0  goes  to  o'\  the  distance  xo"  be- 
ing equal  to  the  hypotlienuse  of  a  triangle  liaving  for  its  base  ox^ 
and  for  its  altitude  o'y^  the  distance  of  tlie  point  from  tlie  plane 
hh.  Then  co"d  is  the  angle  in  its  true  size,  the  bisector  cuts  cd  in 
Sy  and  its  projections  after  the  counter-revohition  are  os^  o's' . 


Fig.  99 


DESCRIPTIVE    GEOMETRY. 


49 


In  Fig.  101,  the  line  OP  is  but  slightly  inclined  to  AB,  so  that 
it  is  inconvenient  to  find  where  it  pierces  either  of  the  principal 


Fig.  102 


planes.  In  this  event,  draw  a  plane  ZZ,  perpendicular  to  AB,  cut- 
ting the  given  lines  in  the  points  P  and  N.  Constructing  the  pro- 
iile,  Fig.  102,  the  vertex  of  the  angle  appears  at  ^,,  and  n^jp^  is  the 
trace  of  the  plane  of  the  two  lines  upon  the  plane  LL.  Revolve 
the  lines  about  this  trace  into  that  plane ;  6>,  goes  to  o'\  the  distance 
xo"  being  equal  to  the  hjpothenuse  of  a  triangle  whose  base  is  o^x^ 
the  distance  of  the  projection  of  the  point  from  the  axis,  and  the 
altitude  is  o'y^  the  distance  of  the  point  from  the  plane  in  which 
the  axis  lies ;   nfi"p^  is  the  angle  in  its  true  size. 

79.  In  Fig.  103,  the  line  OP  is  perpendicular  to  V;  and  so, 
consequently,  is  the  plane  of  the  two  given  lines.  A  supplement- 
ary projection  upon  a  plane  parallel  to  this  plane  at  once  shows  the 
angle  in  its  true  size.  In  this  projection,  the  vertical  plane  will 
appear  (55)  as  a  line  FT^ parallel  to  o'li'^  and  OP  as  a  line  o^p^ 
perpendicular  to  VV.  Draw  any  line  NM  cutting  both  the  given 
lines:  the  points  uj^on  OP  will  all  be  vertically  projected  in  o\ 
and  the  vertical  projection  of  iV^  is  n\  These  points  are  all  pro- 
jected perpendicularly  toward  W^  and  the  distances  of  ??z,,  7i,,  o^^ 
from  VY^  in  the  new  projection,  are  equal  to  the  distances  of  7/z,  n^ 
and  0  from  AB  in  the  horizontal  projection.  The  angle,  being  now 
seen  in  its  true  size,  may  be  bisected  as  before,  the  bisector  cutting 
mi?i,  at  r, :  this  point  is  projected  back  to  7'\  and  thence  to  r,  giv- 
ing 07'  as  the  horizontal  projection  of  the  bisecting  line. 


60 


DESCKIPTIVE    GEOMETRY. 


80.  Such  projections  as  this  are  not  only  of  great  convenience 
in  many  of  tlie  operations  in  abstract  descriptive  geometry,  but  they 
are  of  every-day  occurrence  in  making  mechanical  drawings  for  in- 
dustrial purposes.     Indeed,  the  construction  in  Fig.  103  is  identi- 


Fig.  104 


cal  with  that  used  in  making  the  three  views  of  the  draughtsman's 
triangle,  shown  in  Fig.  104.  Tliis  implement  is  seen  edgewise  in 
the  front  view,  wliere  the  side  CD  is  perpendicular  to  the  paper ;  and 
it  is  foreshortened  in  the  top  view.  Now  in  order  to  exhibit  the 
true  form  and  size,  no  practical  man  will  "  revolve  it  about  its  side 
until  it  becomes  horizontal,"  or  otherwise  disturb  it:  on  the  con- 
trary, leaving  it  at  rest,  a  view  is  drawn,  looking  perpendicularly 
against  it  as  indicated  by  the  arrow.  There  is  no  need  of  using  a 
reference  plane,  like  YY'wl  Fig.  103,  in  ordinary  cases,  because 
its  place  is  supplied  by  lines  or  planes  of  the  object  itself :  thus,  the 
distances  from  EC  in  this  instance  are  the  same  in  the  original  top 
view  and  in  the  third  or  supplementary  view. 

81.  Special  Case  of  the  Above  Problem.  To  find  the  angle 
included  between  the  vertical  and  horizontal  traces  of  a  given  plane  : 
Kevolve  the  vertical  trace  about  the  horizontal  trace  into  H.  In 
Fig.  105,  let  0  be  any  point  on  the  vertical  trace  of  the  plane 
tTt' ;  its  vertical  projection  is  o  on  Tt' ^  and  its  horizontal  projec- 
tion is  o  on  AB.  In  the  revolution,  0  goes  to  o'\  on  a  perpendicu- 
lar to  215   through  o,  the  distance  xo"  being  equal  to  the  hypoth- 


DESCRIPTIVE    GEOMETRY. 


51 


enuse  of  a  triangle  of  which  ox^  oo\  are  the  base  and  the  altitude. 
Also,  To"  is  equal  to  To\  since  the  distance  To'  is  seen  in  its  true 
length,  and  remains  unchanged  during  the  revolution:  and  tTo"  is 
the  required  angle. 


t' 

v^ 

CC/ 

1      \ 

;      r/ 

\    \ 

T/ 

/' 

\\ 

!o  1 

A 

4        Fig.  105 


82.  Problem  10.  To  find  the  angle  between  a  gwen  line  and 
a  given  plane. 

Analysis.  The  angle  which  a  line  makes  with  a  plane  is  the 
same  as  that  included  between  the  line  itself  and  its  projection  on 
the  plane. 

From  any  point  of  the  line,  draw  a  perpendicular  to  the  plane. 
From  any  other  point  of  the  given  line,  draw  a  perpendicular  to 
the  second  line.  This  third  line  will  be  parallel  to  the  projection 
of  the  given  line  upon  the  plane,  and  the  angle  which  it  makes  with 
the  given  line  is  the  one  required. 

Construction.  The  pictoral  representation.  Fig.  106,  illustrates 
the  analysis ;  the  given  line  MN  pierces  the  given  plane  tTt'  at  iT, 


Pig.  106  ^  FiG.  107 

the  perpendicular  from  P  pierces  it  at  (?,  WO  is  the  projection  of 
j^P,  and  PNO  is  the  required  angle.     But  PC^  perpendicular  to 


62 


DESCRIPTIVE    GEOMETRY. 


jP(9,  is  parallel  to  NO^  and  the  angle  PDC  is  equal  to  the  angle 
PNO.  Consequently  it  is  not  necessary  to  find  either  the  point 
iV^or  the  point  O^  but  DC  md^y  be  drawn  anywhere  in  the  project- 
ing plane  determined  by  MN  and  PO. 

Now  in  Fig.  107,  from  any  point  P  on  the  given  line  3IN^ 
draw  a  perpendicular  to  the  given  plane  tlY ;  it  pierces  H  at  7?, 
and  MN  pierces  H  at  E.  Eevolve  the  plane  of  these  two  lines 
into  H  about  its  horizontal  trace  er\  P  goes  to  p" ^  and  rjy'e  is  the 
angle  at  P  in  its  true  size.  From  any  point  d  oiip''e^  draw  dc 
perpendicular  to  p"r^  the  revolved  position  of  the  projecting  line 
PP\  tliQnp'dc  is  the  required  angle. 

Note.  Should  the  given  line  be  inconveniently  situated,  any 
parallel  to  it  may  be  used  instead. 

83.  The  determination  of  the  angle  may,  however,  sonietiiues 
he  facilitated  by  finding  the  actual  projection  of  the  line  upon  the 
plane.     Thus  in  Fig.  108,  the  given  plane  is  parallel  to  AB;   and 


t' 

L 

1 

y 

/f\ 

0 

t 

Fig.  108          ^^ 

t 

p 

Fig.  109 


the  construction  is  as  follows :  Draw  a  plane  ZL  perpendicular  to 
AB;  the  given  line  J/ jY  pierces  this  plane  at  P,  and  the  given 
plane  at  N.  Construct  the  profile.  Fig.  109  :  the  given  plane  is 
here  seen  edgewise  as  the  line  t,t,\  to  which  n'  is  projected  at  n, , 
and  the  point  P  is  found  at  j?,.  From  p,  let  fall  upon  t^t/  the  per- 
pendicular p^o^  ,  and  project  a,  back  to  o' ;  the  distance  of  o  below 
AB  in  Fig.  108  is  equal  to  the  distance  of  o,  in  the  proiile  in  front 
of  V.  "We  have  thus  the  projection  JVO  of  the  line  NP  upon  the 
given  plane;  and  it  is  seen  that  the  line  PO  lies  in  the  plane  ZZ. 


DESCKIPTIVE    GEOMETRY. 


53 


Revolve  N  about  p^o^  into  this  profile  plane ;  N  falls  at  n'\  the  dis- 
tance o,n"  being  equal  to  the  hypothenuse  of  a  triangle  whose  base 
is  o^ti^ ,  and  altitude  n'y ;   and  p{ifi"o^  is  the  angle  sought. 

84.  Special  Cases  of  the  Above  Problem.      In  Fig.  110,  it  is 
required  to  tind  the  angle  made  by  the  plane  tTt'  with  the  ground 


Tig.  110 


^•x? 


Fig.  Ill 


line.  From  any  point  P  on  AB,  draw  Pe^  Pd\  respectively  per- 
pendicular to  Tt  and  Tt' ;  these  are  the  projections  of  a  perpendic- 
ular to  the  plane,  and  0  is  the  point  in  which  it  pierces  the  plane. 
Therefore  TO  is  the  projection  of  AB  upon  tTt\  and  PO  is  per- 
pendicular to  it.  Revolve  the  plane  of  these  two  lines  about  AB 
into  V ;    0  falls  at  o'\  and  PTo'  is  the  required  angle. 

The  points  <?,  d' ^  and  o"  must  lie  upon  the  circumference  of  a 
circle  whose  diameter  is  TP ;  and  the  construction  may  be  facil- 
itated by  drawing  the  circle  first. 

Ill  Fig.  Ill,  the  traces  of  a  plane  tTt\  and  the  projections  of  a 
line  MN^  coincide  in  one  line  inclined  to  AB ;  it  is  required  to  find 
the  angle  between  the  line  and  the  plane.  From  any  point  P  on 
MN^  draw  a  perpendicular  to  the  plane ;  its  projections  coincide, 
therefore  this  perpendicular  cuts  AB  at  R.  Revolve  the  plane  of 
PT  and  PR  about  AB  into  H ;  P  falls  at  j?'',  the  distance  xp"  be- 
ing equal  to  the  hypothenuse  of  a  triangle  whose  base  and  altitude 
are  each  equal  iQ>  px\  draw  Tt*'' perpendicular  iop"P^  1:hGnp"To" 
is  the  required  angle. 

Evidently  o"  is  also  the  revolved  position  of  0^  the  foot  of  a 


54 


DESCRIPTIVE   GEOMETRY. 


perpendicular  let  fall  from  R  to  tlie  plane  tTi\  and  To"  tlie  re- 
volved position  of  tlie  projection  of  AB  upon  that  plane.  There- 
fore RTo"  is  the  angle  made  by  the  ground  line  with  the  given 
plane,  and  RTjp"  is  the  angle  between  AB  and  the  given  line. 

85-  Problem  11.    To  find  the  angle  hetween  two  given  planes. 

Analysis.  Any  plane  perpendicular  to  the  intersection  of  the 
given  planes  will  be  perpendicular  to  both,  and  will  cut  each  of 
them  in  a  line.  These  two  lines  w^ill  be  perpendicular  to  the  inter- 
section at  the  same  point,  and  the  angle  between  tliem  is  the 
required  angle. 

Construction.  This  might  be  accomplished  by  ajij^lying  the 
preceding  problems,  thus:  1.  Find  the  intersection  of  the  given 
planes  as  in  Problem  4.  2.  Through  any  point  of  that  line  draw 
a  plane  perpendicular  to  it,  as  in  Problem  3.  3.  Find  the  inter- 
section of  this  plane  with  each  of  the  given  planes,  as  in  Problem  4. 
4.   Find  the  angle  between  these  two  lines,  as  in  Problem  9. 

But  a  neater  and  less  laborious  process  is  pictorial  ly  represented 
in  Fig.  112,  where  CKD^  CID^  are  the  given  planes,  and  CED  is 
the  liorizontal  projecting  plane  of  CD  their  line  of  intersection. 
The  plane  MFN  is  perpendicular  to  CD^  therefore  its  horizontal 


Fig.  112 


trace  MN  is  perpendicular  to  the  horizontal  projection  CE^  and 
cuts  it  at  0.  MN  is  also  perpendicular  to  the  vertical  line  through 
O^  which  lies  in  the  projecting  plane.  Therefore  MN  is  perpen- 
dicular to  the  plane  CED^  and  consequently  to  OF^  whicli  is  thus 
shown  to  be  the  true  distance  from  0  to  the  vertex  of  the  angle. 


DESCRIPTIVE    GEOMETRY, 


55 


Eevolving  JlfPiT  about  JO^into  H,  P  falls  at  F  on  CE,  OF  he- 
ing  equal  to  OP^  and  MFN  is  the  required  angle. 

86.  This  construction  in  projection  is  given  in  Fig.  113,  where 
tTt\  s.Ss\  are  the  given  planes,  and  CD  is  their  intersection.  Con- 
struct a  supplementary  view,  looking  perpendicularly  against  the 
liorizontal  projecting  plane  of  CI),  as  shown  by  the  arrow ;  in  this 
view  the  horizontal  plane  is  seen  as  M^S^,  in  which  e  lies :  the  al- 
titude dd'  is  tlie  same  as  in  the  vertical  projection,  and  the  line 
CD  is  thus  seen  in  its  true  length  and  inclination.  Draw  ZL  per- 
pendicular to  cd^  in  this  view;  it  is  the  plane  MPJVin  Fig.  112, 
seen  edgewise,  and  cuts  the  horizontal  plane  in  the  line  appearing 
as  7nn,  perpendicular  to  cd,  in  the  horizontal  projection.  It  also 
cuts  cd'  in  a  point  p,  which  may  be  projected  back  to  cd,  thus  de- 
termining j9m,  pn,  the  horizontal  projections  of  the  lines  including 
the  angle  sought.  But  for  the  purpose  of  measuring  the  angle, 
this  is  not  necessary,  since  op  in  the  supplementary  view  is  the  true 
distance  fi-om  0  to  the  vertex;  and  setting  this  distance  off  as  of  on 
cd  in  the  horizontal  projection,  we  have  mfn  as  the  angle  in  its  true 
size. 

In  Fig.  Ill,  the  traces  of  one  plane,  tTt\  are  coincident,  and 
the  other  plane,  sSs\  is  parallel  to  AB.      The  construction  is  the 


Fig.  115 


V^-^  ^  Fig.  114 

same  as  in  Fig.  113,  and  the  two  diagrams  being  lettered  to  corre- 
spond throughout,  no  further  explanation  is  required. 

In  Fig.  115,  both  planes  are  parallel   to  AB;   the  profile  tells 
the  whole  story,  and  the  projections  on  H  and  V  are  simply  useless. 


66 


DESCRIPTIVE   GEOMETRY. 


87.  To  find  the  angle  made  by  a  gUen  plane  with  either  plane 
of  projection. 

In  Fig.  116,  tTf  is  the  given  plane;  to  measure  its  inclina- 
tion to  H,  draw  a  j)lane  sSs^  perpendicular  to  tlie  horizontal  trace, 
as  in  the  diagram  at  the  left,  and  revolve  its  intersection  CD  with 


Fig.  116         ^ 


the  given  plane,  about  cd  into  H ;  D  falls  at  d'\  the  distance  dd'^ 
being  equal  to  dd\  and  dcd'^  is  the  required  angle.  To  find  the 
angle  made  with  the  vertical  plane,  draw  sSs'  perpendicular  to  the 
vertical  trace,  as  in  the  diagram  at  the  right,  and  revolve  the  inter- 
section into  T  about  its  vertical  projection  :  O  falls  at  c'\  c'c"  being 
equal  to  c'c^  and  c'd'c"  is  the  required  angle. 

Conversely  :  Oiven  one  trace  and  the  angle  with  the  corresponding 
plane  of  projection,  to  find  the  other  trace.  This  is  done  by  simply 
reversing  the  preceding  operation.  Thus,  let  \\\ii  horizontal  trace 
and  the  angle  with  H  be  given ;  then  in  the  diagram  at  the  left, 
draw  sS  perpendicular  to  Tt  and  Ss  perpendicular  to  AB :  make 
the  angle  Scg  equal  to  the  given  angle,  draw  at  aS'  a  perpendicular 
to  sS^  cutting  eg  in  d" .  Then  set  up  Sd'  equal  to  8d'\  and  Tt' 
drawn  through  d'  will  be  the  required  vertical  trace. 

88.  Special  Case  of  the  Above  Problem.  In  Fig.  117,  it  is 
required  to  find  the  angle  made  by  the  oblique  plane  tTt'  with  the 
profile  plane  sSs' .  Drawing  the  profile.  Fig.  118,  the  line  of  in- 
tersection CD  is  seen  in  its  true  length  and  inclination  as  cd' ^  and 
tlie  plane  LL  perpendicular  to  it  is  parallel  to  AB.  The  traces  of 
this  plane  in  Fig.  117  are  ZZ,  I'V  \  it  cuts  the  plane  tTt'  in  the  line 
PN.  The  lines  P6>,  PM^  of  Fig.  112,  in  this  case  coincide  in  one 
line,  cut  from  the  profile  plane  s8s'  by  the  plane  LL^  and  seen  in 
its  true  length  as  o'jp^  in  Fig.  118.  Revolve  LL  about  its  vertical 
trace  into  T;   P  falls  at/*,  and  Sfn'  is  the  required  angle. 


DESCRIPTIVE    GEOMETRY. 


57 


Fig.  118  "  Fig.  117 

89.  Problem  12,  To  jhid  the  cominon  perpendicula/r  of  two 
lines  not  in  the  same  plane. 

Analysis.  If  the  two  lines  be  projected  upon  a  plane  parallel 
to  both,  the  pi-ojections  will  be  respectively  parallel  to  the  lines 
themselves,  and  will  intersect  in  a  point.  A  perpendicular  to  the 
plane  at  this  point,  being  perpendicular  to  both  projections  and 
therefore  to  each  line,  will  cut  them  both ;  the  portion  intercepted 
between  them  is  the  required  connnon  perpendicular. 

This  is  illustrated  in  Fig.  119,  where  CD^  MN^  are  the  two 
lines ;  their  projections  cd^  mn,  upon  the  parallel  plane,  intersect  at 
J^ ;  the  intersection  of  their  projecting  planes  is  the  perpendicular 
at  ^,  and  the  intercept  PO  is  the  required  lea^t  distai:ce  between 
the  given  lines.  If  the  plane  approach  the  given  lines,  the  project- 
ing perpendiculars  J/m,  JVn^  will  be  reduced  in  length,  until,  as  in 
Fig.  120,  they  disappear,  and  J/iVlies  in  the  plane,  which  is  paral- 
lel to  CD^  and  its  intersection  with  cd  at  once  determines  the  point 
P.  Upon  this  is  based  one  method  of  construction,  which  consists 
of  tlie  following  steps,  viz.  : 

Construction.  1.  Through  any  point  of  one  line,  draw  a 
])arallel  to  the  other,  and  find  the  traces  of  the  plane  thus  de- 
termined. 

2.  Through  any  point  of  the  second  line,  draw  a  perpendicular 
to  this  plane,  and  find  the  point  in  which  it  pierces  the  plane. 

3.  Through  this  point  draw  a  parallel  to  the  second  line;  it  is 
the  projection  of  that  line  upon  the  plane. 


58 


DESCKIPTIVE    GEOMETRY. 


4.  At  the  intersection  of  this  projection  with  the  first  line, 
erect  a  perpendicular  to  the  plane.  It  will  cut  the  second  line, 
and  is  the  common  perpendicular. 

5.  Determine  the  length  of  the  intercept. 


Fia.l22 
90.  In  the  execution  of  the  above  processes,  the  representa- 
tions  are    all  made    on   the   principal  planes  of  projection.      But 


DESCKIPTIVE   GEOMETRY.  59 

the  result  may  be  attained  in  a  far  more  direct  and  practical  man- 
ner by  tlie  use  of  other  planes.  As  a  preliminary  to  the  explana- 
tion, let  c'd\  Fig.  121,  be  the  vertical  projection  of  a  vertical  line 
in  V ;  its  horizontal  projection  is  o  in  AB :  and  let  MN"  be  an 
oblique  line  in  another  plane.  Let  (?/?,  perpendicular  to  mn,  be 
the  liorizontal  projection  of  a  horizontal  line  intersecting  CD  and 
MN  \  itS' vertical  projection  is  ^'(9' parallel  to  AB,  The  line  P 6^, 
being  horizontal,  is  perpendicular  to  CD ;  and  being  perpendicular 
to  the  horizontal  projecting  plane  of  MN^  it  is  also  perpendicular 
to  that  line ;  moreover,  it  is  seen  in  its  true  length  as  op  in  the 
horizontal  projection. 

91.  Now  in  Fig.  122,  the  group  (1',  2)  contains  the  vertical 
and  the  horizontal  projections  of  two  lines  CD^  MN^  not  in  the 
same  plane;  it  is  required  to  find  their  common  perpendicular- 
Make  a  supplementary  projection,  3',  looking  perpendicularly,  as 
shown  by  the  arrow  a?,  against  the  liorizontal  projecting  plane  Y'  Y' 
of  the  line  CD,  In  this  projection  the  horizontal  plane  appears  as 
H'W  parallel  to  Y'  F',  and  the  distances  of  <?/,  ^/,  above  H'H* 
are  the  same  as  those  of  c' ^  n'^  above  AB  in  the  original  vertical 
projection. 

Now,  retaining  the  same  vertical  plane,  make  a  projection,  4, 
looking  in  a  direction  parallel  to  CD^  as  shown  by  the  arrow  y. 
For  this  purpose  draw  a  new  ground  line,  A^B^ ,  perpendicular  to 
c^d^  \  in  this  projection  CD  will  appear  as  a  point  o^  in  the 
new  ground  line,  and  the  distances  m^r, ,  n^s^^  from  A^B, ,  are 
equal  to  the  distances  mr,  ns^  from  Y'  Y'  in  the  original  horizontal 
projection.  Holding  the  page  so  that  A^B,  is  horizontal,  it  will  be 
perceived  that  the  group  (3',  4)  is  identical  with  Fig.  121 ;  and 
o^jp^ ,  perpendicular  to  m,7i,,  is  the  true  length  of  the  required  line, 
which  is  projected  in  2l  as  o/jf?/.  The  points  ^/,  ji?/,  are  pro- 
jected back  from  3'  to  o,  j?,  in  2,  and  thence  to  o\  p\  in  l',  thus, 
determining  the  projections  of  OP^  the  common  perpendicular,  in 
the  original  position. 

92.  Going  one  step  farther,  make  another  projection,  6',  look 
ing,  as  shown  by  the  arrow  s,  perpendicularly  against  the  horizon- 
tal projecting  plane  m,^,  in  4.      In  this  last  view  the  horizontal 
plane  appears  as  II" H"  parallel  to  m^n^ ;  and  the  distances  above 


60  DESCRIPTIVE    GEOMETRY. 

it,  as  s^n^\  r^m,^^  etc.,  are  the   same  as  the   distances  5,/?/,  Tjn^y 
etc.,  above  A^B^  in  the  vertical  projection  3^ 

In  this  view  tlie  line  CD  appears  as  c^cL^  perpendicular  to 
H"  11" ^  and  both  it  and  the  line  MN  are  seen  in  their  true  lengths. 
What  is  practically  of  equal  importance,  the  angle  between  the  pro- 
jections of  the  two  lines  on  a  plane  parallel  to  both  is  also  seen  in 
its  true  size.  Whereas,  after  executing  the  construction  given  in 
(89),  we  should  be  no  nearer  to  the  determination  of  this  angle 
than  we  were  before ;  and  without  knowing  it,  the  relative  posi- 
tions of  the  tw^o  given  lines  is  not  fully  deiined :  in  fact,  for  prac- 
tical purposes  it  is  absolutely  essential  that  it  should  be  known. 

93.  By  turning  the  page  so  that  the  lines  AB,  H' H\  A,B, , 
H" H'\  in  succession,  are  brought  into  a  horizontal  position,  it 
will  be  perceived  that  each  of  the  groups  (1',  2),  (3',  2),  (3',  4), 
and  (5',  4)  consists  of  a  vertical  and  a  horizontal  projection,  in 
which  are  represented  the  given  and  required  lines,  and  no  others ; 
in  this  respect  also  this  construction  possesses  a  marked  advantage 
over  the  one  first  described. 

The  analysis,  however,  applies  equally  well  to  both;  it  is 
obvious  that  if  a  plane  be  drawn  parallel  to  both  lines,  not  only 
that  plane  but  also  one  of  the  lines  can  be  placed  in  a  vertical  posi- 
tion; and  that,  in  eifect,  is  what  is  done  in  Fig.  122. 

This  has  been  accomplished  by  changing  the  positions  of  the 
planes  of  projection ;  assuming,  1st,  a  new  vertical  plane  contain- 
ing CD ;  2d,  a  new  horizontal  plane  perpendicular  to  CD ;  and  3d, 
another  vertical  plane  containing  MN. 

It  might  have  been  done  by  revolving  the  given  lines,  first^ 
about  a  vertical  axis  until  CD  became  parallel  to  T ;  second^  about 
an  axis  perpendicular  to  V  until  CD  became  vertical ;  and  third ^ 
about  CD  itself  until  MN  became  parallel  to  V.  Both  methods  are 
extensively  used,  and  that  one  should  be  adopted  which  is  best 
suited  to  the  case  in  hand ;  in  the  present  instance,  the  successivo 
rotations  of  the  lines  would  have  resulted  in  a  very  obscure  and  be- 
wildering diagram. 

94.  Problem  13.  To  draw  a  plane  mahing  given  angles  with 
the  princijyal  jplanes. 

Analysis.     This  will  be  best  explained  by  the  aid  of  Fig.  123. 


DESCRIPTIVE   GEOMETRY. 


61 


Suppose  the  problem  solved,  and  let  tTt'  represent  the  plane. 
Through  any  point  P  in  the  ground  line  draw  two  planes;  one 
perpendicular  to  the  horizontal  trace,  cutting  the  plane  tTt'  in  the 
line  MN^  the  other  perpendicular  to  the  vertical  trace,  intersecting 
tT*J  in  CD.  These  lines  intersect  at  O^  and  OP  is  perpendicular 
to  both.  Therefore  in  the  triangle  CPD^  the  angle  GPO  is  equal 
to  the  angle  PDO^  which  is  given;  so  that  if  OP  be  determined, 
the  length  oi  PC  can  be  found,  thus  fixing  the  point  G  in  the  hori- 
zontal trace.     And  OP  can  be  determined  if  PN  is  assumed ;  for 


Fig.  123 


Fig.  124 


since  the  angle  at  M  is  given,  the  triangle  PMN  can  be  con- 
structed, and  OP  is  perpendicular  to  MN,  But  PM  being  per- 
pendicular to  the  horizontal  trace,  tT  must  be  tangent  to  a  circle 
described  in  the  horizontal  plane  about  P  as  a  centre,  with  radius 
PM\  and  Tt'  must  pass  through  the  assumed  point  N. 

Construction.  About  any  point  P  in  AB,  Fig.  124,  describe  a 
circle  with  any  convenient  radius  PM.  Through  M  draw  a  line, 
making  the  angle  PMO  equal  to  the  given  angle  A  with  the  hori- 
'2:ontal  plane,  and  cutting  at  N  the  indefinite  perpendicular  to  AB, 
drawn  through  P.  Draw  PO  perpendicular  to  MN\  make  the 
angle  OPE  equal  to  the  given  angle  v  with  the  vertical  plane,  and 
draw  PE  cutting  MN  in  E.  Set  off  on  the  perpendicular  through 
P^  tlie  distance  PC  equal  to  PE^  and  draw,  through  (7,  tT  tan- 
gent to  the  circle ;  it  is  the  required  horizontal  trace,  cuts  AB  in  T^ 
and  TNt'  is  the  required  vertical  trace. 

95.  Should  the  assigned  value  of  v  be  equal  to  90°  —  A,  the 


p55  DESCRIPTIVE    GEOMETRY. 

intersection  E  will  fall  at  /,  G  will  fall  at  K^  and  tlie  plane  will  be 
parallel  to  AB,  as  shown  in  the  small  profile  at  the  left.  If  v  —  90°, 
as  OPF^  the  intersection  is  at  infinity,  the  horizontal  trace  is  tan- 
gent to  the  circle  at  J!/,  and  the  vertical  trace  coincides  with  MN, 
If  'y  is  greater  than  90°,  as  OPG^  then  GP  produced  cuts  the  pro- 
longation of  iO/  at  ^  below  AB,  and  the  distance  PII  is  to  be  set 
up  as  PZ  above  AB ;  the  horizontal  trace  is  ZSs  tangent  to  the 
circle,  and  the  vertical  trace  (not  shown)  will  pass  through  the 
points  S  and  JV,  If  v  =  90°  -|-  ^)  ^s  OPP^  then  PP  coincides 
with  AB,  its  prolongation  traverses  M,  and  P  W  being  equal  to  the 
radius,  the  horizontal  trace  through  IF,  and  therefore  the  vertical 
trace  through  iV,  will  be  parallel  to  AB,  and  the  position  of  the 
plane  is  shown  in  the  small  profile  at  the  right. 

Should  the  assigned  value  of  v  be  less  than  90°  —  A,  or  greater 
than  90°  4-  A,  it  is  clear  that  the  determining  points  of  intersection 
with  3IJV  will  fall  between  I  and  31 ;  and  since  a  tangent  to  a 
circle  cannot  be  drawn  through  a  point  within  the  circumference, 
the  conditions  cannot  be  satisfied. 

96.  The  principles  and  methods  developed  in  the  discussion  of 
the  preceding  problems  are  sufficient  for  the  solution  of  all  others 
relating  solely  to  the  point,  right  line,  and  plane.  The  geometrical 
principles  involved  are  simple  enough,  but  thorough  mastery  of 
the  manipulations  can  be  acquired  only  by  applying  them  to  a  great 
variety  of  cases ;  the  examples  already  given  illustrating  the  fact 
that  shght  changes  in  the  data  may  cause  the  resulting  constructions 
to  differ  widely  in  appearance. 

A  very  beneficial  exercise  for  those  desii'ous  of  attaining  profi- 
ciency, will  be  found  in  working  the  problems  backward ;  that  is  to 
say,  in  assuming  the  points  and  lines  of  the  construction  to  be  ar- 
ranged in  such  relations  as  may  be  desired,  and  then,  by  reversing 
the  steps  of  the  operation,  ascertaining  the  conditions  which  would 
lead  to  that  development.  Indeed  this  inverted  procedure  is  al- 
most necessary  in  many  cases  to  the  production  of  a  clear  and  ex- 
planatory illustrative  diagram,  and  has  been  used  in  numerous 
instances  in  preparing  the  figures  for  this  volume.  In  practical 
operations,  of  course,  the  conditions  must  be  taken  as  they  are,  let 
the  result  be  what  it  may. 


DESCRIPTIVE   GEOMETRY. 


63 


For  ready  reference,  we  add  here  a  few  applications  and  con- 
structions which  may  subsequently  be  found  useful. 

MISCELLANEOUS    EXAMPLES. 

97.  Example  1.  Given  a  right  line  in  a  given  jplane.  To 
draw  in  that  ;pl(me  another  line  cutting  the  first  in  a  given  jpoint 
at  a  given  angle. 

Construction.  In  Fig.  125,  MN  is  the  given  line,  0  the  given 
point.     Eevolving  the  given  plane  tTt'  about  Tt  into  H,  N  falls  at 


W\  Tn"  being  equal  to  Tn' ;  M  remains  fixed,  and  0  falls  at  o" 
on  mn" .  Draw  Go"d"  making  with  mn"  the  given  angle  co"n" ; 
it  is  the  required  line  in  the  revolved  position  of  the  plane.  In  the 
counter-revolution,  <?,  being  in  the  axis,  remains  fixed,  d"  goes  to 
d' ^  the  distance  Td'  being  equal  to  Td" ;  c'  and  d  are  found  in  AB, 
and  cd^  ed' ^  are  the  projections  of  CD  the  line  required. 

98.  Example  2.  Given  a  right  line  in  a  given  plane.  To 
draw  another  plane  cutting  the  first  in  the  given  line^  and  maTcing 
a  given  angle  with  the  given  plane. 

Construction.  In  Fig.  126,  CD  is  the  given  line  in  the  plane 
tTt' .  Through  any  convenient  point  o  draw  an  indefinite  perpen- 
dicular to  cd^  cutting  Tt  in  m.     Make  a  supplementary  projection 


64 


PESCRIPTIVE    GEOMETRY. 


on  the  horizontal  projecting  plane  of  CD\  the  horizontal  plane 
here  appears  as  II'II\  in  which  c^  lies,  d^d\  is  equal  to  dd\  and 
CD  is  seen  in  its  trne  length  as  c^d^ ;  also,  o  and  m  fall  together  at 
<?,.  Draw  6>,^,  perpendicular  to  c^d^^  and  on  H' II'  set  off  o^g^ 
equal  to  it ;  project  g^  back  to  g  on  cd^  draw  ?7z^,  and  make  the 
angle  nign  equal  to  the  given  angle.  Then  gn  produced  cuts  mo^ 
produced  if  necessary,  in  n^  a  point  in  the  horizontal  trace  required ; 
c  is  a  point  given,  and  scnB  is  the  horizontal  trace,  aSV^'^'  being  the 
vertical  trace,  of  the  second  plane. 

Note,  By  reference  to  Figs.  112  and  113,  it  will  be  perceived 
that  this  operation  is  nearly  the  converse  of  the  process  of  measui*- 
ing  the  angle  between  two  planes.  And  it  should  now  be  readily 
understood  that,  instead  of  making  a  supplementary  view^,  the  hori- 
zontal projecting  plane  of  CD  might  be  revolved  into  either  H  or  Y, 
and  after  the  determination  of  0,^1,  revolved  back  to  its  original 
position. 

99.  Example  3.  To  draw  a  vlane  jparallel  to  a  given  plane y 
at  a  given  distance  front  it. 

Constraction.  In  Fig.  127,  let  tTt'  be  the  given  plane.  Draw 
a  plane  ILL'  perpendicular  to  tT^  and  revolve  it  into  H  about  its- 
horizontal  trace;  cd"  is  the  revolved  position  of  its  intersection 
with  the  given  plane.     Make  cg^  perpendicular  to  Gd'\  equal  to  tlie 


given  distance,  and  draw  e''h  parallel  to  cd" ;  it  is  the  revolved  po- 
sition of  the  line  cut  from  the  plane  ILo'hy  the  required  plane,  and 
cuts  LI  in  ^,  a  point  in  the  horizontal  trace  *aS';  which  is  parallel  to 


DESCRIPTIVE    GEOMETRY. 


65 


iT.  Tlie  vertical  trace  Ss  is  parallel  to  Tt' ;  also,  it  cuts  LV  in 
e\  making  de'  equal  to  de" , 

100.  Example  4.  To  draw  the  projections  of  a  circle  m  a 
given  jjlane. 

Argument.  Let  G  on  the  horizontal  line  CO  in  the  plane  tTt\ 
Fig.  128,  be  the  centre,  about  which  is  to  be  drawn  a  circle  of  a 
given  diameter.  Through  c  draw  an  indefinite  perpendicular  to 
Tt^  and  revolve  the  plane  about  Tt  into  H,  where  C  falls  at  c" ; 
draw  the  circle,  and  circumscribe  it  by  a  square,  whose  sides  are 

1' 


P'     mf 


o'X-^ 

^,yi 

0                     \ 

0^5? 

V 

tX         di 

i    , 

■s 

oVC 

^    / 

^m 

^ 

/ 

Fig.  128 


parallel  and  perpendicular  to  Tt.  At  points  on  the  diameter  <3^"J'', 
as  V\  2'\  draw  ordinates  to  the  circle:  during  tlie  counter-revolu- 
tion, the  lengths  of  a''b"  and  of  its  subdivisions  will  remain  un- 
changed, and  the  sides  of  the  square  which  are  parallel  to  tho  axis 
will  remain  parallel  to  it  and  to  each  other.     So  that  in  the  projec- 


66  DESCRIPTIVE   GEOMETRY. 

tion  the  square  will  appear  as  a  rectangle,  and  the  circle  as  a  curve 
inscribed  within  it ;  and,  since  tlie  radius  and  the  ordinates  which 
are  perpendicular  to  2t  are  foreshortened  in  the  same  proportion, 
it  follows  that  this  curve  is  an  ellipse,  whose  major  axis  ah  is  equal 
to  the  given  diameter  of  the  circle. 

Let  do  and  oo'  be  the  traces  of  a  plane,  perpendicular  to  Tt, 
cutting  the  given  plane  in  the  line  DO',  revolving  this  line  about 
do  into  H,  it  falls  at  do^ :  on  this  set  off  o^e^  =  ac,  then  on  counter- 
revolution e^  falls  at^.  This  gives  oe  as  the  projected  length  of  the 
radius ;  set  oft*  cr  and  cp  equal  to  it,  and  the  length  ^'p  of  the  minor 
axis  of  tlie  ellipse  is  determined. 

101.  Construction.  Draw  through  c  a  parallel  to  Tt,  and  on  it 
set  oft"  ca,  ch,  equal  to  the  given  radius.  Draw  through  e  a  per- 
pendicular to  Tt,  cutting  it  at  ^ ;  on  Tt  set  oft'  grk  equal  to  so',  the 
distance  of  0  from  H ;  draw  Jcc,  produce  it,  and  on  it  set  oft"  e7i, 
em,  equal  to  the  given  radius.  Tlirough  n  and  m  draw  parallels  to 
Tt,  thus  determining  rp  the  minor  axis.  It  is  evident  that  ho  is 
parallel  to  do,  of  the  preceding  argument,  because  oo^  is  equal  to 
oo' ,  and  that  again  to  sc' . 

A  similar  argument  would  apply  to  a  square  circumscribing  the 
circle,  with  its  sides  parallel  and  perpendicular  to  the  vertical  trace. 
Therefore  in  the  vertical  projection  draw  through  c'  a  parallel  to 
that  trace,  and  on  it  set  off  the  major  axis  equal  to  the  given  diam- 
eter. Draw  through  c'  a  perpendicular  to  the  trace,  cutting  it  at 
i ;  set  off  it  equal  to  so,  the  distance  of  C  from  V :  draw  a  line 
through  I  and  c' ,  and  on  it  set  off  c'u,  c'v,  equal  to  the  given  radius. 
Through  u  and  "o  draw  parallels  to  Tt' ,  determining  x  and  y  the 
extremities  of  the  minor  axis.  The  two  ellipses  may  now  be  drawn 
by  any  of  the  usual  methods. 

102.  It  is  obvious  that  p  is  the  horizontal  projection  of  the 
highest  point  in  the  curve,  and  mpq  of  the  tangent  to  the  curve  at 
that  point ;  the  vertical  projection  of  this  tangent  is  g'm!  parallel 
to  AB,  andjt>'  lies  on  g^wl  \  in  like  manner,  ux,  yv,  being  parallel 
to  the  vertical  trace,  will  be  horizontally  projected  as  lines  parallel 
to  AB,  which  will  also  be  tangents  to  the  horizontal  projection  of 
the  circle. 

Moreover,  the  ellipses  will  be  limited  at  the  right  and  left  by 


DESCRIPTIVE   GEOMETRY. 


67 


two  common  tangents  perpendicular  to  AB.  The  exact  location  of 
these  can  be  found  if  desired  by  regarding  each  as  the  intersection 
of  tTt  by  a  proiile  plane.  Draw  through  O  a  plane  perpendicular 
to  AB;  it  cuts  the  given  plane  in  a  line  which,  when  revolved  into 
H  about  Tt.  takes  the  position  o^o" .  Parallel  to  this  draw  two  tan- 
gents to  the  circle,  cutting  Tt  in  f  and  li ;  perpendiculars  to  AB 
thr(jugh  these  points  are  the  tangents  required. 

103.  The  fact  that  in  either  projection,  lines  perpendicular  to 
the  trace  are  all  equally  foreshortened,  while  those  parallel  to  it  are 
not  foreshortened  at  all,  can  be  advantageously  applied  in  many 
other  cases.  Referring  to  the  figure,  it  is  seen  that  in  the  horizon- 
tal projection  the  apparent  length,  cr^  of  the  radius  is  to  the  actual 
length  as  the  base  do  of  the  triangle  doo^  is  to  the  hypothenuse  do^^ 

Now  let  two  scales  be  made,  whose  units  are  to  each  other  in 
this  proportion,  and  divided  into  the  same  number  of  equal  parts; 
then  by  measuring  the  ordinates,  as  V ^  2'',  with  the  larger  scale, 
and  setting  them  off,  as  1,  2,  with  the  smaller  scale,  the  projection 
of  any  plane  curve  can  be  constructed,  often  more  rapidly  than  by 
anv  other  means. 


Fig.  129 


104.  Example  5.   To  revolve  a  given  point  about  a  given  line 


68  DESCRIPTIVE   GEOMETRY. 

which  does  not  contain  the  jpoint^  into  a  given  jplane  which  contains 
neither. 

Analysis.  The  plane  of  rotation  passes  throngli  the  given  point, 
is  perpendicular  to  the  given  line,  and  cuts  it  in  the  centre  of  the 
circular  path.  It  also  cuts  the  given  plane  in  a  line ;  and  if  the 
given  point  reach  the  given  plane  at  all,  it  will  be  at  some  point  of 
this  line. 

Construction.  In  Fig.  129,  7^  is  the  given  point,  J/iYthe  given 
line,  tTt'  the  given  plane.  Draw  through  P  the  plane  sSs  perpen- 
dicular to  MN\  it  cuts  tTt'  in  the  line  DE^  and  MN  in  the  point 
C.  Revolving  the  plane  sBs  a,bout  its  horizontal  trace  into  H,  DE 
falls  in  the  position  de'\  C  falls  at  c"  and  P  at  p" .  About  c" 
describe  a  circular  arc  through  jf)'^ ;  it  cuts  de"  in  the  points  r"  and 
a".  Making  the  counter-revolution,  c"  and  p"  return  to  their 
original  positions,  and  r"  and  o'^  are  found  respectively  at  B  and 
0  on  the  line  DE:  these  are  the  required  points  in  the  giv^en  plane, 
at  one  of  which  P  must  fall  when  revolved  about  the  given  line 
into  that  plane. 


DESCKIPTIVE   GEOMETRY. 


CHAPTER  III. 

Generation  and  Classification-  of  Lines  and  Surfaces.— 
Tangents,  JS^ormals,  and  Asymptotes  to  Lines. — Oscula- 
tion, Eectification,  Radius  of  Curvature. — Tangent,  JS'or- 
MAL,  AND  Asymptotic  Planes  and  Surfaces. 

GENERATION   AND    CLASSIFICATION   OF    LINES. 

105.  Eyery  line   may  be  generated   by   the  motion   of  a  point, 

whicli  is  regarded  as  a  material  particle.  Any  two  successive 
positions  of  the  generating  point,  Laving  no  assignable  distance  be- 
tween them,  are  called  consecutive  points  of  the  line ;  practically, 
they  may  be  considered  as  coincident.  But  in  going  from  one  of 
these  positions  to  the  next,  the  generating  point  moves  in  a  deter- 
minate direction,  whicli  cannot  be  conceived  to  differ  from  that  of 
the  right  line  joining  those  consecutive  points.  This  infinitely  short 
right  line  is  called  an  elementary  line  :  and  every  line  may  be  regarded 
as  made  up  of  an  infinite  number  of  these  rectilinear  elements.  Any 
two  consecutive  elements,  since  they  have  one  point  in  common, 
must  lie  in  one  plane ;  but  they  may  have  different  directions. 

A  Hue  thus  generated  is  called  the  locus  of  the  successive  po- 
sitions of  the  moving  point. 

106.  Lines  are  divided  into  classes,  according  to  the  law  of 
the  motion  of  the  generating  point,  as  follows :  1 .  Right  Lines. 
2,   Single-curved  Lines.      3.   Double-curved  Lines. 

If  the  point  moves  always  in  the  same  direction,  all  the  elements 
lie  in  the  same  direction,  and  the  line  is  a  right  line. 

If  the  point  in  moving  continually  changes  its  direction,  no  two 
consecutive  elements  have  in  general  the  same  direction,  and  the 
line  is  a  curye. 

If  all  the  elements  of  a  curve  lie  in  one  plane,  the  line  is  of 
single  curyature. 


70  DESCRIPTIVE   GEOMETRY. 

If  no  three  consecutive  elements  lie  in  tlie  same  plane,  the  line 
is  of  double  curvature. 

This  last  may  be  more  clearly  seen  by  the  aid  of  Fig.  130,  where 
m,  n,  0,  p,  r,  are  tlie  horizontal,  and  m',  n\  o\  p',  r\  are  the  ver- 
tical, projections  of  five  consecutive  points,  the  elements  MI^^  OP^ 


Fig.  131 


etc.,  being  enormously  magnified.  The  three  points  iT,  (?,  Py 
and  therefore  the  two  elements  NO  and  OP^  lie  in  the  horizontal 
plane,  but  If  lies  above  and  P  lies  below  it.  The  plane  of  MN 
and  NO  is  oblique,  on  being  its  horizontal  trace,  and  it  does  not 
contain  OP :  in  like  manner,  op  is  the  horizontal  trace  of  the  plane 
determined  hj  PP  and  PO,  which  does  not  contain  JVO.  Tims 
it  is  seen  that  any  three,  but  no  four,  consecutive  points — or,  wliat 
is  the  same  thing,  any  two,  but  no  three,  consecutive  elements — - 
lie  in  the  same  plane. 


REPRESENTATION    OF   CURVES. 

107.  A  curve  is  represented  by  its  projections;  which  contain 
the  projections  of  all  its  points,  as  shown  in  Fig.  131.  And  the 
curve  is  in  general  fully  determined  if  its  projections  on  the  princi- 
pal planes  are  given.  For  the  projections  d,  d\  for  instance, 
sufiice  to  locate  the  point  D  in  space  (8),  and  the  same  is  true  of  all 
the  other  points. 

The  traces  of  a  curve  are  found  in  the  same  manner  as  those  of 
a  right  line  (25).  Thus,  in  Fig.  131,  let  the  horizontal  projection 
deg  be  produced  to  cut  the  ground  line  at  Ic ;  then  a  perpendicular 
to  AB  at  that  point,  cutting  the  vertical  projection  in  Tc\  will  lie  iu 
V,  and  ^is  the  vertical  trace  of  the  curve.  ' 


DESCRIPTIVE   GEOMETRY.  71 

JN^o  projection  of  a  double-curved  line  can  in  any  case  be  a  right 
line.  But  if  tlie  plane  of  a  single-curved  line  be  perpendicular  to 
any  plane  of  projection,  the  projections  of  all  points  of  the  curve 
will  lie  in  its  trace,  which  is  a  right  line.  If  the  plane  of  the 
curve  be  parallel  to  any  plane  of  projection,  its  projection  on  tnat 
plane  will  be  similar  and  equal  to  the  curve  itself ;  because  the  pro- 
jection of  each  element  will  be  parallel  and  equal  to  itself  (14).  If 
the  plane  of  the  curve  be  perpendicular  to  the  ground  line,  its  pro- 
jections on  both  the  principal  planes  will  be  right  lines  perpendicu- 
lar to  AB ;  the  curve  is  not  undetermined,  but  is  seen  in  its  true 
form  when  projected  on  a  profile  plane. 

108.  A  curve  may  also  be  generated  by  the  motion  of  a  right 
line  whose  successive  positions  intersect  in  a  series  of  points. 
Thus,  in  Fig.  131^,  the  lines  Z,  Jf,  intersect  at  m;  Jfcuts  iV^at  n^ 


and  so  on.  l^ow  if  the  points  m,  ^,  <?,  ^,  etc.,  are  consecutive, 
the  line  joining  them  will  be  a  curve,  called  the  envelope  of  the 
various  positions  Z,  M^  iV,  etc. ,  of  the  moving  line. 

If  in  this  figure  the  lines  Z,  M^  N,  etc. ,  lie  in  the  plane  of  the 
paper,  the  curve  will  also  lie  in  that  plane.  But  if  they  be  re- 
garded as  the  projections  of  lines  inclined  to  that  plane,  equally  or 
unequally,  it  is  clear  that  the  envelope  will  be  a  line  of  double 
curvature.  In  this  case,  the  lines  Z,  M^  determine  one  plane ;  thQ 
lines  J/,  iV^,  determine  another,  and  JSTis  the  intersection  of  these 
two  planes.  Since  the  like  is  true  of  any  three  of  the  lines  taken 
in  order,  a  double-curved  line  may  also  be  generated  by  the  motion 
of  a  plane  whose  successive  positions  intersect  in  a  series  of  lines. 
These  lines  will  in  general  intersect  each  other  two  and  two  in  con- 


72  DESCRIPTIVE    GEOMETRY. 

secutive  points,  and  tlieir  envelope  will  be  a  line  of  double  curva- 
ture. 


TANGENTS,      NORMALS,      AND      ASYMPTOTES      TO      LINES. OSCULATION, 

RECTIFICATION,   INVOLUTE    AND    EVOLUTE. RADIUS    OF    CURVATURE. 

109.  The  Tangent  to  any  line,  is  a  right  line  drawn  through 
two  of  its  consecutive  points.  Tliese  two  points  determine  the 
direction  of  the  tangent,  and  practically  coincide  in  the  point  of 
contact. 

If  the  given  line  be  of  single  curvature,  the  tangent  will  lie  in 
its  plane ;  for  it  contains  two  points  of  the  curve,  and  they  lie  in 
that  plane.  If  one  right  line  be  tangent  to  another  right  line,  the 
two  lines  wull  coincide;   for  they  have  two  points  in  common. 

Two  curves  are  tangent  to  eacli  other  when  they  have  two  con- 
secutive points  in  common ;  that  is  to  say,  when  they  have  a  com- 
mon tangent  at  a  common  point. 

In  Fig.  IZla^  the  lines  Z,  J/",  iT,  etc.,  will  be  tangents  to  the 
enveloping  curve  when  the  points  m,  n^  <?,  etc.,  become  consecu- 
tive. Suppose  now  a  series  of  curves,  respectively  tangent  at  these 
points  to  the  same  lines ;  then  the  curve  rtmopr  will  be  tangent  to 
them  also.  And  in  fact  the  term  envelope  is  used  in  a  general  sense, 
to  designate  the  line  tangent  to  any  given  series  of  lines,  straight  or 
curved. 

If  two  lines  are  tahgent  to  each  other,  their  projections  on  any 
plane  will  also  be  tangent  to  each  other.  For  the  lines  have  two 
consecutive  points  in  common,  and  the  projections  of  these  will  be 
consecutive  points,  common  to  the  projections  of  the  lines. 

The  converse  of  this  is  not  necessarily  true.  But  if  the  verti- 
cal projections  are  tangent  to  each  other,  and  also  the  horizontal 
projections,  and  the  points  of  contact  are  the  corresponding  projec- 
tions of  a  point  common  to  the  two  lines,  the  lines  themselves  are 
tangent  to  each  other.  Because  the  projecting  perpendiculars  at 
the  common  consecutive  points,  will  determine  two  consecutive 
points  in  space,  whicli  will  be  common  to  the  two  lines. 

The  angle  made  by  a  curve  at  any  point  with  a  plane  or  a  right 
line,  is  the  same  as  that  made  by  its  tangent  at  that  point ;  because 


DESCRIPTIVE    GEOMETRY.  73 

tlie  taiiirent  contains  tlie  rectilinear  element  of  tlie  curve.  Sim- 
ilarlj,  tlie  angle  between  two  curves  is  the  same  as  the  angle  be- 
tw^een  tlieir  tangents  at  the  common  point. 

110.  Osculation.  Osculating  Plane.  A  right  line  can  in  gen- 
eral contain  onlv  two  consecutive  points  of  a  curve.  But  two 
curves  maj  have  three  or  more  consecutive  points  in  common ;  in 
that  case  the  contact,  which  is  more  intimate  than  that  of  simple 
tangency,  is  called  osculation. 

Through  the  tangent  to  a  curve  an  infinite  number  of  planes 
can  be  passed,  and  all  these  planes  are  tangent  to  the  curve.  Each 
will  in  general  contain  only  two  consecutive  points  of  tlie  curve ; 
but  one  of  them  may  be  so  placed  as  to  contain  tkree^  and  this  is 
called  the  osculating  plane.  Thus,  in  Fig.  131«^,  the  line  JT  con- 
tains the  point  m^  the  line  N  contains  the  points  n  and  o ;  therefore 
the  plane  determined  by  M  and  N  passes  through  m^  n^  o.  The 
osculating  plane  of  a  single-curved  line  is,  evidently,  the  plane  of 
the  curve  itself. 

A  Normal  to  any  line  is  a  perpendicular  to  its  tangent,  at  the 
point  of  contact.  An  infinite  number  of  such  perpendiculars  can 
be  drawn,  all  of  which  lie  in  one  normal  plane.  When  not  other- 
wise specified,  '  ^  the  normal ' '  to  any  curve  is  understood  to  be  that 
one  which  lies  in  the  osculating  plane. 

The  rectification  of  a  curve  means  the  determination  of  a  right 
line  equal  to  it  in  length.  If  the  curve  be  so  moved  upon  a  fixed 
tangent  that  the  consecutive  points  of  the  curve  in  their  order, 
come  into  coincidence  with  those  of  the  right  line  in  their  order, 
the  motion  is  one  of  rolling  contact ;  the  curve  measures  itself  off 
upon  the  tangent,  and  the  part  rolled  over  is  equal  in  length  to  the 
arc  of  the  curve  whicli  has  rolled  over  it. 

111.  Involute  and  Evolute.  If  the  tangent  roll  upon  a  fixed 
curve,  any  point  of  it  will  describe  a  second  curve,  w^iich  is  called 
an  involute  of  the  first ;  and  the  fixed  curve  is  called  the  evolute  of 
the  second. 

In  Fig.  131«^,  suppose  a  thread,  wound  upon  the  broken  line 
knr^  to  be  kept  taut,  and  unwound  as  indicated  by  the  arrow. 
Then  it  is  clear  that  the  path  of  5,  a  point  upon  the  position  Z,  will 
be  made  up  of  a  series  of  circular  arcs,  st^  tu^  etc. ,  of  whicli  the 


74  DESCRIPTIVE    GEOMETRY. 

centres  are  tlie  points  m,  n,  o,  etc.  When  these  points  become 
consecutive,  the  change  of  centres  and  radii  goes  on  continuously, 
the  broken  hue  becomes  the  evolute,  to  which  the  positions  of  the 
thread,  Z,  M^  iT,  etc.,  are  all  tangent;  and  the  path  svx  becomes 
an  involute  of  Tcnr,  But  the  nature  of  the  motion  is  not  changed ; 
the  tracing  point  at  any  instant  is  describing  a  circle  about  the 
point  of  contact,  and  therefore  moving  in  a  direction  per];>endicular 
to  the  radius.  Consequently,  the  tangent  to  the  evolute  is  always 
normal  to  the  involute. 

What  is  true- of  the  motion  of  s  is  equally  true  of  the  motion 
of  any. other  point/*,  situated  upon  the  same  lineZ;  whose  path 
fgh  also  becomes  ultimately  an  involute  of  the  curve  knr.  Hence 
a  single-curved  evolute  may  have  an  infinite  number  of  involutes 
in  its  own  piano.  These  are  not  in  general  similar  curves ;  but 
they  SiVQ  parallel^  in  the  sense  that  the  normal  distance  between 
any  two  of  them  is  constant.  On  the  other  hand,  since  the  inter- 
sections of  consecutive  normals  to  any  curve  determine  an  evolute, 
a  single-curved  line  can  have  but  one  evolute  in  its  own  plane. 
And  this,  being  the  envelope  of  all  the  normals  to  the  given  curve, 
whether  consecutive  or  not,  can  be  drawn  with  much  precision,  if 
the  curve  is  such  that  the  direction  of  its  tangent,  at  any  point 
assumed  at  pleasure,  can  be  determined. 

112.  Centre  of  Curvature.  In  Fig.  131a,  let  7io,  o-p,  be  two 
consecutive  elements  of  a  curve;  .these,  if  produced,  form  two 
successive  tangents  iV,  0,  whose  included  angle,  called  the  angle  of 
contingence,  is  a  measure  of  the  rate  of  curvature  at  o.  Draw  a 
perpendicular  to  each  of  these  elements  at  its  middle  point ;  these 
perpendiculars  intersect  at  <?,  the  centre  of  a  circle  passing  through 
n,  (?,  and^. 

At  n  and  ^,  draw  perpendiculars  to  iV,  and  at  o  and  ^,  per- 
pendiculars to  0 ;  the  second  pair  of  perpendiculars  will  cut  the 
first  pair  at  the  points  d^  e^  and  the  three  angles  at  ^,  c,  ^,  will  be 
equal  to  each  other  and  to  the  angle  of  contingence.  All  these 
perpendiculars  are  understood  to  lie  in  the  osculating  plane.  Now 
in  the  case  of  the  actual  curve  here  represented  by  a  broken  line, 
the  points  d  and  e  will  coincide  with  c ;  for  the  distances  on^  op^ 
being  infinitely  small,  the  three  perpendiculars  to  N  will  form  one 


DESCRIPTIVE    GEOMETRY.  75 

single  line,  and  tlie  three  perpendiculars  to  0  will  form  another. 
The  angle  of  contingence  is  also  infinitely  small ;  nevertheless  the 
point  6',  then  the  intersection  of  two  consecutive  normals,  is  in  gen- 
eral at  a  finite  distance  from  ^,  and  is  the  centre  of  a  circle  whose 
circumference  contains  the  three  consecutive  points  n^  (?,  ^,  of  the 
curve. 

This,  then,  is  the  osculating  circle ;  and  since  its  circumference 
has  the  same  rate  of  curvature  as  that  of  the  curve  itself  at  <9,  its 
centre  c  is  called  the  centre  of  curvature,  and  go  is  called  the  radius 
of  curvature,  of  the  given  curve  at  that  point. 

The  intersections  of  successive  normals  to  any  curve  will  be  a 
series  of  consecutive  points,  determining  in  general  another  curve, 
which  is  the  locus  of  the  centre  of  curvature.  In  the  case  of  a 
])lane  curve^  this  locus  will  be  the  evolute  of  the  given  curve. 
Thus,  in  Fig.  IZla^  iV^and  6>,  consecutive  normals  to  svx,  meet  at 
o  on  the  curve  hnr^  and  o  is  the  centre  of  curvature  at  v.  This 
does  not  hold  true  of  double-curved  lines. 

113.  An  Asymptote  to  a  line  is  another  line  which  the  given 
line,  during  a  portion  of  its  course,  continually  approaches,  becom- 
ing tangent  to  it  only  when  its  own  length  becomes  infinite. 

Ordinarily,  the  asymptote  to  a  plane  curve  is  a  right  line,  which 
becomes  tangent  to  the  curve  at  an  infinite  distance.  But  this  is 
not  essential :  for  example,  the  evolute  of  the  Archimedean  spiral 
has  a  circular  asymptote  of  finite  diameter,  and  the  curve  lies 
wholly  within  the  circumference ;  again,  two  conjugate  hyperbolas, 
having  common  rectilinear  asymptotes,  are  asymptotic  to  each  other. 
Also,  a  right  line  may  be  an  asymptote  to  a  line  of  double  cur- 
vature. 


GENERATION    AND    CLASSIFICATION    OF    SURFACES. 

114.   Every  surface  may  be  generated  by  the  motion  of  a   line. 

This  moving  line  is  called  the  generatrix,  and  its  difiierent  positions 
are  called  elements,  of  the  surface.  Any  two  successive  positions 
of  the  generatrix,  having  no  assignable  distance  between  tliem,  are 
called  consecutive  elements ;  practically  they  may  be  regarded  as 
coinciding. 


76  DESCRIPTIVE    GEOMETRY. 

Surfaces  may  be  separated  into  two  grand  divisions,  according 
to  the  form  of  the  generatrix,  viz,  : 

I.   Ruled  Surfaces,  which  contain  rectihnear  elements. 

II.  Double-Curved  Surfaces,  which  have  no  rectilinear 
elements. 

In  other  words,  the  surfaces  of  the  first  division  can  be  gener- 
ated by  the  motion  of  right  lines,  while  those  of  the  second  division 
cannot.  The  former  may  also  be  generated  by  the  motion  of  curved 
lines ;  the  latter  cannot  be  generated  without  it. 

115.  A  right  line  may  move  so  that  all  its  positions  lie  in  the 
same  plane.  It  may  move  otherwise ;  in  which  case  any  two  con- 
secutive positions  either  will  lie  in  the  same  plane,  or  they  w^ill  not. 
According  to  the  law  of  the  motion  of  the  rectilinear  generatrix, 
then,  ruled  surfaces  are  subdivided  into  three  classes,  as  follows : 

1 .  Plane  Surfaces All  the  rectilinear  elements  lie  in 

the  same  plane. 

2.  Single-curved  Surfaces. .Any  two  consecutive  rectilinear 
elements  lie  in  the  same  plane. 

3.  Warped  Surfaces No  two  consecutive  rectilinear  ele- 
ments lie  in  the  same  plane. 

plane  surfaces. 

116.  The  Plane  Surface  is  unique.  That  is  to  say,  there  is 
but  one  form  of  plane,  and  there  neither  are  nor  can  be  any  differ- 
ent kinds.     All  planes  are  flat,  and  one  is  no  flatter  than  another. 

The  rectilinear  generatrix  may  move  so  as  to  toach  another 
right  line,  remaining  always  parallel  to  its  first  position ;  so  as  to 
touch  two  other  right  lines  which  are  parallel  to  each  other,  or 
which  intersect;  or  it  may  revolve  about  another  right  line  to 
which  it  is  perpendicular. 

Acquaintance  with  the  nature  and  properties  of  planes  was 
necessarily  assumed  at  the  outset;  tlie  methods  of  representing 
them,  and  of  assuming  points  and  lines  in  them,  have  already  been 
described. 


DESCRIPTIVE   GEOMETRY.  77 


SURFACES   OF    SINGLE    CURVATURE. 

117.  Single- curved  surfaces  are  of  three  varieties,  viz. : 

1.  Cones In  which  all  the  rectilinear  elements  intersect  in 

a  common  point. 

2.  Cylinders  . .  In  which  all  the  rectilinear  elements  are  parallel 
to  each  other. 

3.  Con  volutes..  In  which  the  consecutive  elements  intersect 
two  and  two,  no  three  having  a  common  point. 

CONICAL    surfaces. 

118.  In  generating  a  cone,  the  right  line  moves  so  as  always 
to  touch  a  given  curve  called  tlie  directrix,  and  also  to  traverse  a 
given  jDoint  called  the  vertex.  Since  the  generatrix  is  indefinite  in 
length,  the  surface  is  divided  at  the  vertex  into  two  parts,  called 
respectively  the  upper  and  lower  nappes.  It  is  clear  that  in  the  case 
of  a  given  cone,  any  line  drawn  upon  the  surface  so  as  to  cut  all 
tlie  elements  may  be  taken  as  a  directrix,  and  any  element  as  the 
generatrix. 

The  cone  may  also  be  generated  by  the  motion  of  a  curve  which 
always  touches  a  given  right  line,  and  changes  its  size  according  to 
a  proj^er  law. 

119.  The  portion  of  the  cone  usually  considered,  is  included 
between  tlie  vertex  and  a  plane  which  cuts  all  the  elements;  the 
curve  of  intersection  is  called  the  base,  and  its  form  gives  a  distin- 
guisliing  name  to  the  surface — as  a  cone  with  a  circular,  a  para- 
bolic, an  elliptical,  or  a  spiral  base,  as  the  case  may  be.  If  the 
base  has  a  centre,  a  right  line  drawn  through  this  centre  and  the 
vertex  is  called  the  axis  of  the  cone.  The  point  in  which  any  ele- 
ment pierces  the  plane  of  the  base  is  called  the  foot  of  the  element. 

A  definite  portion  of  either  nappe,  in«.Juded  between  two  paral- 
lel planes  which  cut  all  the  elements,  is  called  a  frustum  of  the 
cone,  the  limiting  curves  being  called  respectively  the  upper  and 
lower  bases. 

A  secant  plane  through  the  vertex  cuts  the  cone  in  rectilinear 
elements  intersecting  the  base. 


'8 


DESCKIPTIVE    GEOMETRY. 


120.  A  ri^ht  cone  is  one  all  of  whose  rectilinear  elements  make 
equal  angles  with  a  right  line  passing  through  the  vertex,  which  is 
called  the  axis.  This  is  also  called  a  cone  of  reyolution,  since  it 
can  be  generated  by  revolving  the  hypothenuse  of  a  right-angled 
triangle  about  one  of  its  sides  as  an  axis. 

If  the  directrix  of  a  cone  be  changed  to  a  right  line,  or  if  the 
vertex  be  placed  in  the  plane  of  a  single- curved  directrix,  the  cone 
will  degenerate  into  a  plane. 

If  the  vertex  be  removed  to  an  infinite  distance,  the  rectilinear 
elements  will  be  parallel  to  each  other,  and  the  cone  will  become  a 
cylinder. 

121.  Representation  of  the  cone.  A  cone  is  represented  by  the 
projections  of  the  vertex,  one  of  the  curves  of  the  surface  (usually 
its  plane  base),  and  the  principal  rectilinear  elements.  Thus,  in 
Fig.  132,  let  O  he  the  vertex;  draw  mxny,  the  horizontal  projec- 


riG.132 


ria.133 


tion  of  the  base,  and  ox,  oy,  tangent  to  that  curve :  this  completes 
the  horizontal  projection  of  the  cone.  The  j^lane  of  the  base  in 
this  case  is  perpendicular  to  V ;  its  vertical  projection  is  therefore  Vj 
right  line,  limited  at  m'  and  n'  by  tangents  to  the  horizontal  pro- 
jection perpendicular  to  AB;  the  vertical  projections  m'o\  n'o\  of 
the  extreme  visible  elements,  complete  the  representation  of  the 
surface. 

To  assume  a  rectilinear  element,  assume  a  point  on  the  curve  of 
the  base,  as  C  or  D,  and  draw  through  it  a  right  line  to  the  vertex. 


DESCRIPTIVE   GEOMETRY.  79 

To  assume  a  point  on  the  surface,  assume  one  of  its  projections, 
say  the  horizontal,  as  jp ;  througli  jp  draw  the  horizontal  projection 
of  an  element,  o'p ;  this  element  intersects  the  base  at  2>,  and  ^' 
must  lie  on  d'o\  the  vertical  projection  of  the  element.  The  hori- 
zontal projecting  plane  oi  DO  cuts  the  cone  in  another  element, 
{7(9,  having  the  same  horizontal  projection ;  upon  this  lies  another 
point  7?,  whose  horizontal  projection  r  coincides  with  p. 

122.  Particular  attention  is  called  to  the  fact  that  the  cone  ic 
here  shown  with  its  base  not  situated  in  the  horizontal  plane. 
And  it  is  so  shown  for  the  purpose  of  illustrating  and  emphasizing 
another  fact,  viz.,  tliat  the  projection  of  the  base  must  not  be  con- 
founded with  the  base  itself.  It  is  very  natural  and  proper  to  place 
the  base  of  a  cone  or  a  cylinder  upon  the  horizontal  plane ;  but  if 
it  be  always  so  placed,  experience  has  proved  that  the  above  neces- 
sary distinction  is  very  apt  to  be  lost  sight  of,  wdiich  may  lead  to 
serious  errors  in  the  subsequent  applications  of  problems  relating  to 
these  surfaces. 

CYIJNDRICAL    SURFACES. 

123.  The  Cylinder,  as  intimated  in  (120),  is  merely  that  limit- 
ing form  of  the  cone  in  which  the  vertex  is  infinitely  remote ;  and 
it  may  be  generated  by  a  right  line  Avhich  moves  so  as  always  to 
touch  a  given  curve,  and  have  all  its  positions  parallel.  In  the 
case  of  a  given  cylinder,  any  line  of  the  surface  which  cuts  all  the 
rectilinear  elements  may  be  taken  as  a  directrix,  and  any  one  of 
those  elements  as  the  generatrix. 

A  cylinder  may  also  be  generated  by  a  curvilinear  generatrix, 
all  of  whose  points  move  in  the  same  direction  and  with  the  same 
velocity. 

124.  A  plane  cutting  all  the  rectilinear  elements,  intersects  the 
cylinder  in  a  curve  called  its  base ;  whose  form,  as  in  the  case  of 
the  cone,  gives  a  distinguishing  name  to  the  cylinder.  If  the  base 
has  a  centre,  a  right  line  drawn  through  the  centre,  parallel  to  the 
elements,  is  called  the  axis. 

If  a  definite  portion  of  the  surface  included  between  two  paral- 
lel planes  is  considered,  the  two  curves  of  intersection  are  called 
the  upper  and  lower  bases. 


80  DESCRIPTIVE    GEOMETRY. 

A  plane  parallel  to  the  rectilinear  generatrix  cuts  the  cylinder, 
if  at  all,  in  rectilinear  elements  intersecting  the  base. 

125.  A  ri^ht  cylinder  is  one  whose  rectilinear  elements  are  per* 
pendicular  to  the  plane  of  the  base ;  and  the  base  itself  is  then  said 
to  be  a  right  section. 

A  right  cylinder  with  a  circular  base  is  also  called  a  cylinder  of 
revolution,  since  it  may  be  generated  by  revolving  one  side  of  a  rec- 
tangle about  the  opposite  side  as  an  axis. 

If  the  curvilinear  directrix  of  any  cylinder  be  changed  to  a 
right  line,  the  surface  will  degenerate  into  a  plane. 

126.  The  projecting  lines  of  the  various  points  of  a  curve,  as 
seen  in  Fig.  131,  are  rectilinear  elements  of  a  riglit  cylinder,  whose 
base,  in  the  plane  of  projection,  is  the  projection  of  the  curve. 
Thus  the  curve  in  space  is  determined  by  the  intersection  of  two 
cylinders,  called  respectively  the  horizontal  and  vertical  projecting 
cylinders. 

127.  The  representation  of  the  cylinder  differs  from  that  of  the 
cone  only  in  this,  that  the  projections  of  the  rectilinear  elements 
are  parallel  instead  of  convergent.  If  a  limited  portion  is  to  be  rep- 
resented, the  projections  of  both  bases  must  be  drawn;  if  not,  the 
projections  of  the  extreme  vibible  elements  may  terminate  indefi- 
nitely, as  shown  in  Fig.  133. 

To  assnme  a  rectilinear  element,  assume  a  point  on  the  curve  of 
the  base,  as  C  or  Z^,  and  draw  through  it  a  right  line  parallel  to 
the  rectilinear  generatrix. 

To  assume  a  point  on  the  surface,  assume  one  of  its  projections, 
say  the  vertical,  as  p  ;  through  this  draw  the  corresponding  pro- 
jection o'p'  of  an  element ;  this  element  intersects  the  base  at  2>, 
and  p  must  lie  on  do^  the  horizontal  projection  of  the  element.  Tlie 
vertical  projecting  plane  oi  DO  cuts  the  cylinder  in  another  ele- 
ment, CK^  having  the  same  vertical  projection ;  and  upon  this  ele- 
ment lies  another  point,  7?,  of  the  surface,  whose  vertical  projec- 
tion coincides  w^itli  that  of  P. 

CONVOLUTE    SURFACES. 

128.  The  Convolute  may  be  generated  by  a  right  line  which 
moves  so  as  always  to  be  tangent  to  a  line  of  double  curvature.   Any 


DESCRIPTIVE   GEOMETRY. 


81 


two  consecutive  rectilinear  elements,  but  no  three,  will  lie  in  the  same 
plane ;  for  thej  are  the  extensions  of  the  elements  of  the  directrix, 
of  which  (106)  any  two,  but  no  three,  consecutive  ones  intersect 
each  other. 

Suppose  a  piece  of  paper  cut  in  the  form  of  a  right-angled  tri- 
angle to  be  wrapped  around  a  regular  polygonal  prism,  Fig.  134:, 
its  base  becoming  the  perimeter  of  the  base  of  the  prism.  Then 
the  hypothenuse  will  become  a  broken  line,  each  portion  lying  in  a 


Fig.  136 


}.131 


face  of  the  prism  and  being  equally  inclined  to  its  edges.  In  un- 
winding, the  paper  turns  about  each  edge  in  succession  as  upon  a 
hinge,  until  it  coincides  with  the  plane  of  the  next  face ;  when  the 
free  portion  of  the  hypothenuse  will  be  an  extension  of,  and  there- 
fore tangent  to,  the  element  of  the  broken  line  which  lies  in  that 
face.  And,  considering  the  successive  positions  of  the  hypothenuse, 
it  is  seen  that  No.  1  intersects  No.  2  at  7>,  'No.  2  intersects  Xo.  3 
at  ^,  No.  3  cuts  No.  4  at  J^,  and  so  on ;  but  No.  1  does  not  inter- 
sect No.  3,  nor  does  No.  2  intersect  No.  4.  Now  let  the  sides  of 
the  polygon  be  indelinitely  increased  in  number ;  the  base  will  ul- 
timately become  a  circle,  the  prism  will  become  a  cylinder ;  tlte 
broken  line  will  become  a  helix,  and  its  tangents,  then  consecutive, 


82  DESCRIPTIVE    GEOMETRY. 

will  lie  in  a  continuous  surface,  of  wliicli  a  limited  portion  is  sliown 
in  Fig.  135. 

The  point  M  of  the  unwinding  paper  will  always  lie  in  the  curve 
myn  in  the  horizontal  plane ;  this  curve,  which  is  the  involute  of 
ih^:^  circular  base,  is  the  horizontal  trace  of  the  surface,  which,  ex- 
panding as  it  rises,  winds  around  the  cylinder  in  convolutions  like 
those  of  a  sea-shell.  The  cylinder  itself  has  no  connection  with  the 
surface ;  it  is  introduced  merely  in  order  to  throw  the  nearer  por- 
tion of  the  convolute  into  stronger  relief. 

As  in  the  cases  of  the  cone  and  the  cylinder,  the  curve  of  inter- 
section with  any  plane  which  cuts  all  the  rectilinear  elements  may 
be  taken  as  the  base  of  the  convolute. 

129.  The  Edge  of  Kegression.  The  generatnx,  in  Fig.  135,  is 
shown  as  limited  in  length ;  thus,  the  elements  J/0,  NG^  termi- 
nate at  O  and  ^,  their  points  of  contact  with  the  directrix. 

But  they  may  be  continued  past  those  points,  as  indicated  in 
d<>tted  lines ;  and  their  extensions,  6^7?  and  GII^  lie  in  a  continua- 
tion of  the  surface,  which  expands  in  other  successive  whorls :  and 
these  two  portions,  or  nappes,  of  the  convolute  have  in  common 
the  helical  directrix. 

But  it  is  to  be  noted  that  this  is  not  a  curye  of  intersection ;  for, 
as  shown  in  Fig.  130,  wdiich  represents  a  detached  portion  of  the 
surface  on  a  larger  scale,  the  rectilinear  elements  which  lie  upon 
the  lower  nappe  neither  pierce  the  upper  nappe  nor  cut  the  helix. 
Indeed,  they  are  tangent  to  the  helix  by  hypothesis ;  and  the  sur- 
face is  continuous  and  unbroken,  although  reflected  sharj^ly  upon 
itself,  and  forming  at  the  helix  what  is  called  an  edge  of  regression. 
This  is  a  limiting  line,  at  which  a  surface  terminates  abruptly  by 
the  law  of  its  generation :  it  is  always  formed  by  the  intersection 
of  consecutive  generating  lines,  whether  they  are  right  lines,  as  in 
this  case,  or  curved  ones. 

A  surface  may  also  be  reflected,  or  bent  back,  in  an  analogous 
manner,  along  a  line  which  is  not  thus  formed ;  the  limiting  line  in 
that  case  is  called  a  gorge  line. 

Since  there  is  an  infinite  number  of  double-curv^ed  lines,  a  great 
variety  of  convolutes  may  also  exist,  with  peculiarities  depending 
upon  those  of  their  directrices ;   but  the  one  above  described  will 


DESCRIPTIVE   GEOMETRY.  83 

suffice  for  illustration.  It  has  been  selected  for  the  reasons  that  it 
is  not  only  as  simple  as  any,  but  possesses  some  interesting  proper- 
ties, which  will  be  noticed  in  due  course,  and  render  it  practically 
more  important  than  others.  The  methods  of  constructing  and  rep- 
resenting it  will  also  be  subsequently  discussed  in  connection  with 
problems  relating  to  it. 

GENERATION    OF    SINGLE-CURVED    SURFACES    BY    MOVING    PLANES. 

130.  Observing  that  any  two  consecutive  elements  of  the  con- 
volute determine  an  osculating  plane  of  the  directrix  (110),  and 
that  each  element  is  the  intersection  of  two  successive  positions  of 
the  osculating  plane  (108),  it  will  now  be  seen  that  this  surface 
may  be  generated  by  the  motion  of  a  plane  subject  to  the  condition 
that  it  shall  always  be  oscillatory  to  a  given  line  of  double  curva- 
ture. Since  three  points  not  in  one  right  line  suffice  to  locate  a 
plane,  this  single  condition  will  in  general  control  absolutely  the 
motion  of  the  plane  generatrix,  and  determine  the  form  of  the 
resulting  surface.  An  exceptional  case  occurs  when  the  directrix 
is  reduced  to  a  point,  the  motion  of  the  plane  being,  then  indeter- 
minate  unless  governed  by  another  condition,  which  may  be  de- 
duced as  follows : 

If,  on  any  single-curved  surface,  any  curve  be  drawn  which 
cuts  all  the  rectilinear  elements,  any  two  consecutive  ones  will  inter- 
cept an  element  of  that  curve ;  the  plane  of  those  two  elements  will 
therefore  be  tangent  to  the  curve  (110).  In  the  case  just  men- 
tioned the  surface  becomes  a  cone;  which  rnay,  consequently,  be 
generated  by  the  motion  of  a  plane  which  always  passes  through  a 
given  point  and  also  remains  tangent  to  a  given  curve.  If  the  vertex 
be  infinitely  remote,  the  cone  will  become  a  cylinder ;  all  of  whose 
elements,  being  parallel,  are  perpendicular  to  one  and  the  same 
plane.  Hence  a  cylinder  may  be  generated  by  the  motion  of  a 
plane  which  is  always  tangent  to  a  given  curve  and  also  always  per- 
pendicular to  a  given  plane. 

WARPED    SURFACES. 

131.  The  absolute  motion  of  a  right  line  in  space  is  fully  deter- 
mined when  the  simultaneous  motions  of  any  two  of  its  pomts  are 


84  DESCRIPTIVE    GEOMETRY. 

given  in  direction  and  Telocity ;  the  form  of  the  surface  generated  by 
the  moving  hne  will  be  determined  if  the  directions  and  relative 
velocities  of  these  two  motions  be  known. 

These  directions  may  be  determined  by  reqniring  that  the  rec- 
tilinear generatrix  shall  always  touch  two  other  given  lines,  either 
straight  or  curved ;  but  some  third  condition  is  necessary  in  order 
to  esta])lisli  a  definite  ratio  between  the  velocities. 

By  whatever  means  this  is  accomplished,  it  is  clear  that  if  upon 
the  resulting  surface  any  other  line  be  drawn  which  cuts  all  the 
rectilinear  elements,  that  line  may  be  taken  as  a  third  directrix; 
and  the  same  surface  will  be  produced  if  tlie  rectilinear  generatrix 
move  so  as  always  to  touch  this  last  line  and  the  two  at  first 
assumed. 

Any  ruled  surface  whatever,  then,  may  be  generated  by  a  right 
line  so  moving  as  always  to  touch  three  given  linear  directrices. 

132.  Cone  Directer  and  Plane  Directer.  Suppose  the  three 
directrices  to  be  so  chosen  that  tlie  resulting  ruled  surface  is  neither 
plane  nor  single-curved ;  then  through  any  given  point  in  space  let 
a  series  of  consecutive  right  lines  be  drawn,  parallel  in  tlieir  order 
to  the  consecutive  rectilinear  elements  of  the  given  warped  surface 
in  their  order.  These  w^ill  be  elements  of  a  cone,  called  the  cone 
directer  of  the  surface :  and  it  is  clear  that  if  any  two  lines  which 
cut  all  the  rectilinear  elements  be  taken  as  directrices,  a  right  line 
moving  so  as  always  to  touch  those  lines  and  have  its  consecutive 
positions  parallel  to  the  consecutive  elements  of  the  cone  directer, 
will  re-generate  the  same  surface. 

It  is  easy  to  see  that  the  warped  surface  may  be  such  that  the 
series  of  lines,  drawn  through  the  assumed  point  parallel  to  its  ele- 
ments, shall  lie  in  one  plane — which  is  a  limiting  form  of  the  cone ; 
and  this  plane  is  called  the  plane  directer. 

Any  warped  surface  whatever,  therefore,  may  be  generated  by  a 
right  line  moving  so  as  always  to  touch  two  given  lines,  and  have  its 
consecutive  positions  parallel  either  to  a  given  plane  directer,  or  to 
the<  consecutive  elements  of  a  given  cone  directer. 

133.  Since  every  warped  surface  is  curved,  it  is  possible  also  to 
conceive  it  as  being  generated  by  a  curve,  .which  moves,  and  at  the 
same  time  changes  its  form,  according  to  some  definite  law. 


DESCRIPTIVE   GEOMETRY.  85 

There  is  a  great  variety  of  warped  surfaces,  with  peculiarities 
depending  on  the  laws  of  their  formation.  The  methods  of  repre- 
senting them,  of  assuming  points  and  lines  upon  them,  etc. ,  require 
for  ready  apprehension  a  familiarity  with  some  matters  not  yet  dis- 
cussed, and  are  accordingly  reserved  for  subsequent  consideration. 

DOUBLE -CURVED    SURFACES. 

134.  A  Double-curved  Surface  is  one  which  contains  no  rectilin- 
ear elements,  and  can  be  generated  only  by  a  curve  which  moves  in 
such  a  manner  as  not  to  generate  a  surface  of  either  of  the  preced- 
ing classes. 

Double-curved  surfaces  may  be  either  double  convex,  that  is, 
convex  in  all  directions,  as  the  surface  of  a  sphere  or  an  egg ;  or 
concavo-convex,  that  is,  convex  in  some  directions  but  concave  in 
others,  as  the  surface  of  a  bell  or  of  the  groove  in  a  pulley.  Both 
these  peculiarities  may  exist  in  a  single  unbroken  surface,  as  in  the 
case  of  a  cylindrical  ring,  or  annular  torus. 

SURFACES    OF    REVOLUTION. 

135.  A  Surface  of  Revolution  is  one  which  may  be  generated  by 
the  revolution  of  a  given  line  about  a  right  line  as  an  axis. 

Tlie  intersection  of  such  a  surface  by  a  plane  perpendicular  to 
the  axis  is,  therefore,  the  circumference  of  a  circle.  Consequently 
the  surface  may  also  be  generated  by  a  circle  which,  moving  with 
its  centre  in  the  axis  and  its  plane  perpendicular  to  it,  changes  its 
radius  according  to  a  definite  law. 

This  second  mode  of  generation  is  the  one  to  which  the  greater 
pi-actieal  interest  attaches ;  since  it  is  in  this  manner  that  such  sur- 
faces, of  extensive  application  in  the  mechanic  arts,  are  actually 
produced  in  the  lathe. 

A  plane  traversing  the  axis  is  called  a  meridian  plane,  and  its 
intersection  with  the  surface  is  called  a  meridian  line :  all  meridian 
lines  of  the  same  surface  are  obviously  identical,  and  any  one  of 
them  may  be  taken  as  the  revolving  generatrix. 

136.  If  the  revolving  line  be  straight,  it  either  will  lie  in  the 
same  plane  with  the  axis,  or  it  will  not.     If  it  does,  it  will  either 


86 


DESCRIPTEVE   GEOMETRY. 


be  parallel  to  it,  or  intersect  it ;  in  the  former  case  the  surface  will 
be  a  cylinder,  in  the  latter  a  cone ;  and  these  are  the  only  sin^le- 
cnryed  surfaces  of  reyolution. 

If  the  revolving  right  line  does  not  lie  in  the  same  plane  witli 
the  axis,  then,  1 :  Every  point  of  it  moves,  therefore  the  consecu- 
tive positions  do  not  intersect ;  and,  2  :  Every  point  more  remote 
from  the  axis  moves  faster  than  one  nearer  to  it,  therefore  the  con- 
secutive positions  are  not  parallel.  The  surface  must,  then,  be 
warped ;  its  meridian  line,  as  will  subsequently  appear,  is  an  hy- 
perbola :  and  this  is  the  only  warped  surface  of  reyolution.  It  may 
also  be  generated  by  revolving  an  hyperbola  about  its  conjugate 
axis ;  and  the  surface  being  unbroken,  it  is  also  known  as  the  liyjper- 
holoid  of  revolution  of  one  nappe. 

With  the  exception  of  the  three  just  considered,  all  surfaces  of 
revolution  are  of  double  curvature. 

137.  If  two  surfaces  of  revolution,  having  a  common  axis,  cut 
or  touch  each  other  at  any  point,  they  will  do  so  all  around  the  cir- 
cumference of  the  circle  described  by  that  point.  Thus  in  Fig. 
137,  the  meridian  lines  macn,  oacp^  are  tangent  to  each  otlier  at  a^ 


"Y 

Y 

f 

\ 

T 

>_ 

cj 

\ 

'/ 

p 


Fig.  137 


and  intersect  each  other  at  c ;  they  will  maintain  these  relations 
throughout  the  revolution,  and  the  circles  ab^  cd^  will  be  common 
to  the  two  surfaces. 

138.  Representation  of  Surfaces  of  Revolution.  These  surfaces 
are  represented  by  drawing  two  views,  viz.  :  1,  a  side  view,  show- 
ing the  meridian  contour,  and  should  this  be  a  broken  line,  such 


DESCRIPTIVE    GEOMETRY.  87 

circles  as  are  described  by  the  intersections ;  and  2,  an  end  view,  in 
wliicli  are  drawn  tlie  largest  circle  of  the  surface,  and  such  others  as 
niaj  be  necessary  in  order  that  the  drawings  may  be  clear  and 
easily  read. 

For  all  ordinary  practical  purposes,  and  for  most  of  the  pur- 
poses of  descriptive  geometry  as  well,  any  reference  to  a  ground 
line  is  useless,  if  not  worse.'  As  shown  in  Fig.  138,  a  line,  con- 
tinuous centre  line,  7d,  containing  the  axis,  should  be  drawn 
through  and  heyond  both  views ;  another  one,  gh^  should  be  drawn 
at  right  angles  to  the  first,  through  the  centre  of  the  end  view : 
since  these  lines  are  imaginary,  they  should  never  terminate  in  any 
outline,  lest  they  be  supposed  to  represent  lines  actually  existing  on 
the  surface. 

The  axis,  when  there  is  no  reason  to  the  contrary,  is  supposed 
to  be  parallel  to  the  paper,  and  either  liorizontal  or  vertical  as  may 
be  more  convenient ;  in  the  latter  case  the  end  view  is  a  "  liorizon- 
tal projection  " — but  whether  placed  above  or  below  the  side  view, 
this  horizontal  projection  is  invariably  a  top  view,  and  represents 
the  object  as  seen  from  ahove^  never  as  seen  from  below. 

139.  To  assume  a  point  on  the  surface,  assume  one  of  its  pro- 
jections, for  instance  c  in  the  end  view.'  Then  the  point  nmst  lie 
on  the  surface  of  a  cylinder  whose  radius  oc  is  known ;  draw  at  that 
distance  from  the  axis  a  parallel  to  it,  in  the  side  view ;  this  is  the 
outline  of  the  cylinder,  and  cuts  the  meridian  line  in  m  and  n ; 
these  points  describe  circles,  to  one  or  both  of  which  c  is  projected^ 
as  at  c'  or  c^' . 

If  the  projection  in  the  side  view  be  assumed,  as  at  d\  then 
the  point  lies  on  a  circle  whose  radius  pr  can  be  found.  With  thi& 
radius  describe  an  arc  about  o  in  the  end  view ;  the  other  projec- 
tion must  lie  on  this  arc,  as  at  d  or  d^, 

TANGENT,     NORMAL,    AND    ASYMPTOTIC    PLANES    AND    SURFACES. 

140.  If  on  any  surface,  any  number  of  lines  be  drawn  through 
a  given  point,  then  the  tangents  to  all  these  lines  at  the  common 
point  will  in  general  lie  in  one  and  the  same  plane. 

Such  a  plane  is  said  to  be  tangent  to  the  surface ;  and  the  point 
is  called  the  point  of  contact. 


88  DESCRIPTIVE    GEOMETRY. 

If  the  surface  is  a  single- curved  one,  no  plane  can  be  tangent 
to  it  at  anj  point,  which  does  not  coincide  with  some  position  of 
the  plane  generatrix.  That  generatrix  contains  two  consecutive 
rectilinear  elements^  and  is  tangent  (130)  to  every  curve  of  the 
surface  which  cuts  them  both.  An  infinite  number  of  such  curves 
can  be  drawn  through  any  point  of  either ;  hence  the  plane  is  tan- 
gent all  along  the  line  in  which  those  two  elements  practically  coin- 
cide. 

In  the  case  of  a  double-convex  surface,  the  demonstration  is  as 
follows:   Let  M^  Fig.  138(3^,  be  a  section  of  the  surface  by  any 


plane,  and  T  a  tangent  to  it  at  any  point  jp ;  if  the  plane  be  re- 
volved about  T  as  an  axis,  this  line  will  also  be  tangent  to  any  suc- 
cessive section,  as  iT  or  0\  for  it  contains  two  consecutive  points 
of  iH/,  and  they  remain  fixed  during  the  revolution.  Through  ]) 
draw  on  the  surface  any  other  curves,  as  K^  Z,  cutting  M  2>X  «,  h. 
The  section  N  cuts  these  curves  at  a' ^  V  \  the  section  0  cuts  them 
at  a" ^  y ;  therefore  the  secants  ^a,  jpb ;  jpa\  jpV ;  jpa" ^  ph'%  al- 
w^ays  lie  in  the  revolving  plane.  Ultimately,  the  section  of  the 
surface  will  cut  K  and  L  at  points  consecutive  to  j?,  and  these 
secants  will  become  tangents,  still  lying  in  the  same  plane  through 
T.  If  the  surface  is  concavo-convex,  whether  warped  or  double- 
curved,  the  same  argument  applies,  although  the  forms  of  the  sec- 
tions J/,  iV^,  etc. ,  wull  be  different. 

To  pass  a  plane  tangent  to  any  surface  at  a  given  point,  there- 
fore :  Draw  through  the  point  any  two  intersecting  lines  of  the 
surface,  and  at  the  point  draw  a  tangent  to  each  line ;  the  plane 
determined  by  these  two  tangents  is  the  plane  required. 

141.  A  plane  tangent  to  any  ruled   surface    must  in  general 


DESCRIPTIVE   GEOMETRY.  89 

contain  the  rectilinear  elements  which  pass  through  the  point  of 
contact;  for  each  rectilinear  element  is  its  own  tangent  (109),  and 
therefore  lies  in  the  tangent  plane  by  the  preceding  definition. 
The  vertex  of  a  cone  is  an  exceptional  case ;  through  that  point  an 
infinite  number  of  planes  tangent  to  the  surface  may  be  passed, 
each  of  which  contains  two  consecutive  rectilinear  elements,  but  no 
others. 

Since  a  plane  tangent  to  any  single-curved  surface  is  tangent  all 
along  a  rectilinear  element  (140),  it  follows  that  if  the  base  of  the 
surface  lie  in  any  plane  of  projection,  the  corresponding  trace  of 
the  tangent  plane  will  be  tangent  to  the  base,  at  the  point  in  which 
the  element  of  contact  pierces  that  plane  of  projection.  And  in  any 
case,  the  right  line  cut  from  the  plane  of  the  base  by  the  tangent 
plane  will  be  tangent  to  the  curve  of  the  base. 

142.  A  plane  tangent  to  a  warped  surface  contains  the  rectilinear 
element  which  passes  through  the  point  of  contact ;  but  since  it 
does  not  contain  the  consecutive  one,  it  cannot  in  general  be  tan- 
gent along  the  element ;  but  there  are  some  cases  in  which  it  is. 
If  the  surface  has  two  sets  of  rectilinear  elements,  the  tangent 
plane  will  contain  both  those  which  pass  through  the  point  of 
contact. 

A  plane  containing  one  rectilinear  element  of  a  warped  surface, 
And  not  parallel  to  the  consecutive  ones,  will  cut  each  of  them  in  a 
point.  The  curve  joining  these  points  will  cut  the  given  element, 
and  the  given  plane  will  be  tangent  to  the  surface  at  the  point  of 
intersection ;  for  it  contains  the  given  element,  which  is  its  own 
tangent,  and  also  the  tangent  to  the  curve  of  intersection  at  the 
point  mentioned.  If  the  given  plane  be  parallel  to  a  plane  directer 
of  the  surface,  there  will  be  no  such  curve,  and  the  plane  will  not 
in  general  be  tangent  to  the  surface. 

Consequently,  unless  the  projecting  planes  of  the  rectilinear 
elements  are  parallel  to  a  plane  directer,  each  of  them  will  be  tan- 
gent to  the  surface  at  some  point.  The  projecting  lines  of  these 
points  form  the  projecting  cylinder  of  the  surface ;  and  the  pro- 
jections of  the  elements,  being  the  traces  of  the  projecting  planes, 
will  all  be  tangent  to  the  base  of  this  cylinder,  which  lies  in  the. 
plane  of  projection. 


90  DESCRIPTIVE   GEOMETRY. 

143.  A  plane  tangent  to  a  surface  of  reyolution  is  perpendicular 
to  the  meridian  plane  passing  through  the  point  of  contact.  Be- 
cause, it  contains  the  tangent  to  the  circle  of  the  surface  at  that 
po^nt ;  and  this  tangent,  lying  in  a  plane  perpendicular  to  the  axis, 
is  perpendicular  to  the  radius  at  its  extremity,  and  also  to  any  line 
joining  that  extremity  with  the  axis ;  and  both  this  radius  and  that 
line  lie  in  the  meridian  plane. 

144.  Two  curved  surfaces  are  tangent  to  each  other  when  they 
have,  at  a  common  point,  a  common  tangent  plane.  Evidently, 
the  sections  of  the  two  surfaces  made  by  any  one  plane  passing 
through  the  point  of  contact  will  be  tangent  to  each  other  at  that 
point. 

If  two  surfaces  of  revolution  having  a  common  axis  are  tangent 
to  each  other  at  a  point,  they  will  be  tangent  all  round  tlie  circum- 
ference of  the  circle  described  by  that  point  in  the  generation  of 
the  surfaces  (137). 

If  two  single- curved  surfaces  are  tangent  to  each  other  at  a 
point  of  a  common  rectilinear  element,  they  will  be  tangent  all 
along  that  element.  Because  the  plane  tangent  to  either  surface  at 
any  point  is  tangent  to  it  all  along  the  rectilinear  element  passing 
through  that  point  (140). 

It  is  possible  also  to  construct  two  warped  surfaces  which  shall 
be  tangent  to  each  other  all  along  a  common  rectilinear  element. 
They  must  then  have,  at  every  point  of  that  element,  a  common 
tangent  plane ;  the  methods  by  which  this  condition  can  be  satisfied 
will  be  explained  farther  on. 

145.  Normal  Lines,  Planes,  and  Surfaces.  A  right  line  is  normal 
to  a  surface  at  any  point  when  it  is  perpendicular  to  tlie  tangent 
plane  at  that  point. 

A  curbed  line  is  normal  to  a  surface  at  any  point  when  its  tan- 
gent at  that  point  is  normal  to  the  surface.  "When  not  otherwise 
stated,  ''the  normal"  to  a  surface  is  understood  to  be  rectilinear. 

A  plane  is  normal  to  a  surface  at  any  point  when  it  is  perpen- 
dicular to  the  tangent  plane  at  that  point.  Thus  there  may  be  an 
infinite  number  of  planes  normal  to  the  surface  at  a  point,  while 
there  can  be  but  one  normal  right  line,  common  to  all  these  planes. 

If  at  the  consecutive  points  of  any  line  upon  a  given  surface '  a 


DESCRIPTIVE    GEOMETRY.  91 

series  of  normals  to  that  surface  be  erected,  they  will  be  elements 
of  a  ruled  surface  normal  to  the  giyen  surface.  If  these  normals  are 
tangent  at  those  consecutive  points  to  lines  lying  upon  any  other 
surface,  then  that  surface  is  also  normal  to  the  given  surface. 
This  relation  is  mutual,  and  the  two  surfaces  are  said  to  be  normal 
to  each  other. 

146.  Asymptotic  Planes  and  Surfaces.  The  relation  between 
asymptotic  surfaces  is  analogous  to  that  between  asymptotic  lines. 
Thus  if  an  hyperbola  be  made  the  base  of  a  right  cylinder,  the 
plane  containing  its  asymptote  and  parallel  to  the  elements  of  the 
cylinder  will  be  asymptotic  to  the  surfacCc  Again,  if  two  conju- 
gate hyperbolas,  together  with  their  common  asymptotes,  be  re- 
volved about  either  axis,  the  two  hyperboloids  of  revolution  will  be 
asymptotic  to  each  other  and  to  the  cone  generated  by  the  asymp- 
totes. 


DESCHIPTIVE   GEOMETEY. 


CHAPTER   ly. 

On  the   Determination  of   Planes  Tangent   to    Surfaces   of 
Single  and  of  Double  Curvature. 

planes  tangent  to  single-curved  surfaces. 

147.  In  the  construction  of  these  problems,  and  of  many  others, 
there  is  frequent  occasion  to  draw  tangents  to  curves  of  unknown 
properties.  A  sufficient  degree  of  accuracy  for  j^resent  purposes, 
and  indeed  for  most  practical  purposes,  may  be  ol)tained  by  means 
of  approximating  circular  arcs,  as  follows :  Let  it  be  required  to 
draw  a  tangent  to  the  curve  DJ^,  Fig.  139,  at  the  point  P.  By 
Q  trial  and  error  a  centre  O  and  radius 

CP  can  be  found,  sucli  that  the  cir- 
cular arc  described  about  0  will 
sensibly  coincide  with  the  given 
curve  for  a  short  distance  on  each 
Tig.  139  gj^^  ^f  ^]^q  given  point ;   the  required 

tangent  is  then  drawn,  perpendicular  to  CP. 

If,  on  the  other  hand,  it  be  required  to  draw  a  tangent  in  a  given 
direction,  or  through  a  given  point,  as  0 :  In  tliis  case  the  tangent 
is  drawn  mechanically ;  the  ruler  being  set,  not  so  as  to  coincide 
with  either  the  point  or  the  curve,  but  at  a  small  distance  from 
each,  the  eye  being  able  to  judge  with  perfect  precision  as  to  the 
equality  of  these  distances.  The  point  of  contact  is  then  deter- 
mined by  dropping  a  perpendicular  upon  the  tangent  from  the 
centre  O,  found  as  aboye. 

147a.  The  following  constructions  are  more  laborious,  but 
give  more  precise  determinations : 

1.  In  Fig.  139<^,  to  draw  a  tangent  to  the  curve  A^Z  at  any 
pointy.  Through  ^  draw  chords  from  any  points  of  the  curve, 
produce  them  all  in  one  direction  (say  to  the  left),  then  with  j?  as 


DESCRIPTIVE    GEOMETRY. 


93 


a  centre  and  any  convenient  radius  draw  a  circular  arc  C^F"  inter- 
secting them.  P>om  tins  arc  set  off  on  the  prolongation  of  each 
chord  a  distance  equal  to  the  chord  itself,  the  chords  on  the  left  of 
^  being  set  oft"  to  the  left  of  the  arc,  and  vice  versa;  thus  ef  =  pi, 


.V        Fig.  139  a 


cd  =^  pa^  ns  ^=  pg^  and   so   on.     The  cnvYQ  fds  thus  determined 
cuts  6^  F  in  a  point  o  of  the  required  tangent  TT. 

2.  In  Fig.  189 J,  to  iind  the  point  of  contact,  TT  being  tan- 
gent to  the  curve  KL.  Draw  any 
number  of  chords  parallel  to  TT^ 
through  their  extremities  draw 
parallel  ordinates  in  opposite  direc- 
tions, and  on  each  ordinate  set  off 
from  TT  a  distance  equal  to  the 
corresponding  chord ;  as,  for  ox- 
ample,  rs  —  mn  =  gh^  hd  ^^  ef  =^ 
ac^  etc.  ;  the  curve  sdn^  passing 
through  the  points  just  located, 
cuts  TT  in  the  required  point. 

Such    curves    are    known    as 
"  curves  of  error,"  and  with  due  care  give  very  accurate  results. 

148.  Problem  1.  To  draw  a  plane  tangent  to  any  single- 
curved  surface  through  any  given  point  of  the  surface.  ' 

Analysis.  Through  the  given  point  draw  a  rectilinear  element, 
and  through  the  foot  of  that  element  draw  a  tangent  to  the  base. 
The  plane  of  these  two  lines  is  the  required  tangent  plane. 

Construction.  In  Fig.  140,  the  given  surface  is  a  cone  whose 
vertex  is  O^  the  plane  of  the  base  being  perpendicular  to  T ;  this  is 
represented,  and  the  point  P  upon  it  assumed,  as  in  Fig.  132. 
Through  P  draw  the  element  OPC^  and  through  its  foot  6^  draw  a 


94 


DESCRIPTIVE    GEOMETRY. 


tangent  to  the  base.  The  horizontal  projection  he  of  this  tangent 
will  be  tangent  at  c  to  the  horizontal  projection  of  the  base,  and  its 
vertical  projection  h'c'  will  coincide  with  the  vertical  projection  of 
the  base.  The  traces  of  the  tangent  are  M  and  G ;  those  of  the 
element  are  N  and  D ;  therefore  m!  and  n'  are  points  in  the  verti- 
cal trace,  and  d  and  g  are  points  in  the  horizontal  trace,  of  the  re- 
quired plane ;  and  these  traces  must  meet  at  T  in  the  ground  line. 
In  Fig.  141,  the  surface  is  a  cylinder,  with  its  base  in  the  ver- 
tical plane.     The  tangent  to  the  base  is  therefore  the  required  ver- 


^       Fig.  142 


tical  trace,  which  cuts  AB  at  T.     The  element  PC  pierces  H  in  Z>, 
thus  determining  dTt  the  horizontal  trace. 

In  Fig.  142,  the  base  of  the  cylinder  is  horizontal ;  the  liori- 
zontal  trace  is  therefore  parallel  to  lic^  the  horizontal  projection  of 
the  tangent  to  the  base.  The  point  cZ,  found  as  before,  fixes  the 
location  of  this  trace,  which,  when  produced,  cuts  AB  in  T.  The 
tangent  KC  pierces  V  in  the  point  N\  and  since  the  vei-tical  trace 
of  the  element  PC  \s>  inaccessible,  the  direction  of  Tt'  is  determined 
by  drawing  it  through  n'. 


DESCRIPTIVE    GEOMETRY. 


95 


149.  Problem  2.  To  draw  a  plane  tangent  to  a  cone  through 
a  given  point  without  the  surface. 

First  Method.  Analysis.  Draw  a  line  through  the  vertex  of 
the  cone  and  the  given  point.  Through  the  point  in  which  this 
line  pierces  the  plane  of  the  cone's  base,  draw  a  tangent  to  the 
base.      The  plane  of  these  two  lines  is  the  required  tangent  plane. 

Construction.  Let  0-XY^  Fig.  143,  be  the  given  cone,  P  the 
given  point.  The  line  OP^  through  the  point  and  the  vertex, 
pierces  H  in  D^  and  the  plane  of  the  cone's  base  in  S.     The  verti- 


cal projection  of  the  tangent  is  s'c'  tangent  to  the  vertical  projec- 
tion of  the  base ;  its  horizontal  projection  coincides  with  that  of 
tlie  base  itself,  whose  plane  is  perpendicular  to  H.  This  tangent 
pierces  H  in  6^ ;  and  <7  and  g  are  two  points  in  the  horizontal  trace 
of  the  required  plane,  which  cuts  AB  in  T,  one  point  of  the  verti- 
cal trace.  Another  point  might  be  determined  by  finding  the  ver- 
tical trace  of  OP ;  but  in  this  instance  a  third  line,  the  element  of 
contact  C?^',  has  been  employed  instead;  it  pierces  T  in  ^,  and 
Tr't'  is  the  required  vertical  trace. 


96  DESCRIPTIVE    GEOMETRY. 

When  the  cone  becomes  a  cylinder,  as  in  Fig.  144,  the  line 
through  the  vertex  becomes  parallel  to  the  elements.  In  the  dia- 
gram, the  plane  of  the  base  is  parallel  to  T,  and  OP  pierces  it  at 
8\  in  this  particular  case  the  vertical  projection  s'c'  of  the  tangent 
to  the  base  happens  to  be  ]3arallel  to  AB,  and  since  SO  is  therefore 
parallel  to  the  horizontal  trace,  the  required  plane  will  be  parallel 
to  the  ground  line ;  consequently  it  is  sufficient  to  determine  one 
point  in  each  trace. 

When  more  than  one  tangent  to  the  base  can  be  drawn,  there 
will  be  more  than  one  solution. 

150.  Second  Method.  Analysis.  Pass  through  the  given  point 
any  plane  cutting  all  the  rectilinear  elements  of  the  given  surface, 
and  in  tins  plane  draw  through  the  point  a  tangent  to  the  curve  of 
intersection.  Draw  through  the  point  of  contact  a  ]-ectilinear  ele- 
ment ;  this  element,  and  the  tangent  to  the  curve,  determine  the 
required  plane. 

Note. — This  process  may  be  applied  to  any  single  curved  sur- 
face ;  but  is  of  more  particular  advantage  in  cases  analogous  to  the 
one  herewith  illustrated. 

Construction.  In  Fig.  145  the  surface  is  a  cone,  with  a  circular 
base  situated  in  the  horizontal  plane :  every  section  of  it  by  a  hori- 
zontal plane  will  therefore  be  a  circle  whose  centre  lies  upon  the 
line  0  U^  drawn  from  the  vertex  to  the  centre  of  the  base.  Through 
the  given  point  P  draw  a  horizontal  plane  LL ;  it  cuts  0  U  in  7?, 
and  also  cuts  tlie  element  OX  in  a  point  whose  vertical  projection 
is  s' ;  therefore  r's'  is  the  radius  of  the  circular  section,  which  more- 
over must  be  tangent  to  the  extreme  visible  elements  in  the  hori- 
zontal projection.  Draw  through  P  a  tangent  to  this  section  ;  in 
this  case  this  tangent  is  parallel  to  the  horizontal  trace,  which  is 
taugent  to  the  base  of  the  cone ;  so  that  it  is  not  necessary  to  make 
use  of  the  element  of  contact,  OG.  But,  especially  if  the  given 
point  P  is  but  a  small  distance  above  H,  it  would  be  advisable  to 
use  the  vertical  trace  of  that  element  if  accessible,  since  the  direc- 
tion of  Tt'  would  be  thus  more  accurately  determined. 

151.  Problem  3.  To  draw  a  plane  tangent  to  a  given  cone 
and  parallel  to  a  given  right  line. 

Analysis.     Through  the  vertex  of  the  cone  draw  a  parallel  to  the 


DESCRIPTIVE    GEOMETRY.  97 

given  line ;  and  from  the  point  in  which  it  pierces  the  plane  of  the 
cone's  base,  draw  a  tangent  to  the  base.  Tlie  plane  of  this  tangent 
and  the  parallel,  is  the  tangent  plane  required. 

Construction.  If  in  Fig.  143  we  supj)ose  the  line  OP  to  be 
determined  by  the  condition  that  it  shall  be  parallel  to  a  given  line 
J/jy,  the  construction  is  the  same  as  in  (149). 

If  the  parallel  line  through  the  vertex  pierces  the  plane  of  the 
base  in  a  point  so  situated  that  a  tangent  to  the  base  cannot  be 
drawn  through  it,  the  problem  is  impossible ;  if  more  than  one  tan- 
gent can  be  drawn,  there  will  be  a  corresponding  number  of  solu- 
tions. 

If  in  this  or  the  preceding  problem  tlie  line  through  the  vertex 
be  parallel  to  the  plane  of  the  base,  the  tangent  to  the  base  will  be 
parallel  to  that  line.  Should  that  line  pierce  the  plane  of  tlie  base 
at  a  remote  and  inaccessible  point,  the  tangent  plane  may  be  con- 
structed as  follows :  Through  any  point  of  the  parallel  line  pass  a 
plane  cutting  all  the  rectilinear  elements ;  and  from  the  assumed 
point  draw  a  tangent  to  the  curve  of  intersection.  This  tangent, 
and  the  rectilinear  element  through  the  point  of  contact,  will  de- 
termine the  required  plane. 

152.  Wlien  the  vertex  is  infinitely  remote,  a  process  analogous 
to  that  of  (150)  may  be  employed;  in  any  plane  containing  the 
given  line  and  cutting  all  the  rectilinear  elements  of  the  cylinder,  a 
tangent  to  the  curve  of  intersection  may  be  drawn  parallel  to  the 
given  line :  the  plane  of  this  tangent  and  the  element  through  the 
l^oint  of  contact  will  be  the  required  tangent  plane.  This  is  illus- 
trated in  Fig.  146,  PC  being  the  tangent  and  PE  the  element; 
and  obviously  the  intersection  CE  of  the  tangent  plane  with  the 
plane  of  the  base  will  be  tangent  to  the  base  at  E^  the  foot  of  the 
element  of  contact. 

The  direct  application  of  this  process  is  not  usually  convenient ; 
but  from  ii  a  tentative  one  is  derived,  based  upon  the  consideration 
that  if  any  plane  DOF\>Q  constructed,  containing  a  line  (97^  paral- 
lel to  the  elements,  and  another  line  DO  parallel  to  the  given  line : 
then  the  intersection  7)7^  with  the  plane  of  the  base  of  the  cylinder 
will  be  parallel  to  CE^  which  line,  with  the  element  through  the 
point  of  contact  E,  will  determine  the  required  plane. 


98 


DESCRIPTIVE    GEOMETRY. 


153.  The  construction  in  accordance  with  the  preceding  argu- 
ment is  shown  in  Fig.  147,  where  JO^is  tlie  given  line.  Through 
any  point  0  on  any  element  of  the  cylinder,  as  OX,  draw  a  parallel 
to  JfiV^;  this  line  pierces  the  plane  of  the  base  at  6^,  the  element 
pierces  it  at  X,  and  XC  is  the  line  of  intersection  corresponding  to 
-Z>i^of  Fig.  146.     Parallel  to  X6' draw  a  tangent  to  the  base;  it 


riG.146 


pierces  T  in  /^,  H  in  G^  and  E  is  the  point  of  contact.  The  ele- 
ment through  ^pierces  H  in  i>  and  V  in  R :  joining  d  and  ^,  then, 
we  have  the  horizontal  trace ;  joining  s^  and  r\  the  vertical  trace  is 
determined ;  and  these  traces  meet  at  T  in  the  ground  line. 

154.  Special  Cases  of  the  Preceding  Problems.  In  Fig. 
14S  a  cone  of  revolution  is  given,  with  its  axis  parallel  to  AB :  it  is 
required  to  draw  a  plane  tangent  to  it,  parallel  to  the  given  line 
LM.  The  parallel  through  the  vertex  0  pierces  W  in  Z>,  V  in  iT, 
^nd  the  plane  of  the  cone's  base  in  S.  In  the  profile,  the  circle  of 
the  base  is  seen  in  its  true  form,  and  S>  is  projected  at  6-, :  the  tan- 
gent to  the  base  through  «,  pierces  H  at  ^,,  whose  distance  from 
FF  determines  the  distance  of  g  from  AB  in  tlie  horizontal  projec- 
tion. Then  the  points  d  and  g  fix  the  horizontal  trace  tT^  and 
Tn't'  is  the  vertical  tr^ce.  If  necessary  or  convenient,  the  element 
of  contact,  seen  in  the  profile  as  o^c^^  and  in  the  vertical  projection 
-as  o'c\  may  be  used  to  locate  points  in  the  traces ;  for  example,  the 
point  /''  in  the  vertical  trace  is  found  by  producing  that  element. 

In  the  profile.  Fig.  149,  a  cylinder  of  revolution  is  shown,  its 


DESCRIPTIVE   GEOMETRY. 


99 


axis  O  being  parallel  to  the  ground  line.  This  figure  sufficiently 
illustrates  the  fact  that  in  this  case,  whether  the  plane  is  to  be  tan- 
gent at  a  given  point  P  on  the  surface,  to  pass  through  a  given 
point  R  without  the  surface,  or  to  be  parallel  to  a  given  line  LM^ 
the  solution  is  at  once  effected  and  the  result  clearly  exhibited  by 
drawing  the  profile.  The  projections  on  the  principal  planes  are 
not  here  given ;  as  in  Fig.  74,  they  would  be  made  up  chiefly  of  a 


^ 

^ 

[- 

Xb---'-/^^'^^ 

•^ 

V 

\X 

0 

;■ 

\ 

^  \ 

'\ 

\, 

Fig.  149 

> 

V 

confusing  series  of  parallels :  and  all  of  this  is  equally  true  whether 
the  base  of  the  cylinder  is  circular  or  of  any  other  form. 

155.  PiANES  Tangent  to  the  Helical  Convolute.  In 
Fig.  150  is  shown  so  much  of  this  particular  convolute  as  is  neces- 
sary to  illustrate  the  application  to  it  of  the  preceding  processes. 

in  representing  it,  the  helical  directrix  should  be  accurately 
drawn ;  and  this  is  facilitated  by  first  drawing  the  front  and  top 
views  of  the  cylinder  upon  whose  surface  it  lies.  Now,  the  axis  of  ; 
this  cylinder  being  vertical  and  its  base  lying  in  H,  it  was  shown 
in  (128)  tliat  the  horizontal  trace  of  the  convolute  will  be  the  in- 
volute of  the  circle  of  that  base ;  and  this  should  next  be  carefully 
cjonstructed,  since  it  is  the  locus  of  the  points  in  which  H  is  pierced 
fy  the  rectihnear  elements  of  the  surface.  Then,  om  being  equal 
to  the  quadrant  oli^  and  gn  being  three  times  as  great,  it  is  clear 
that  OM  is  tangent  to  the  helix  at  0^  and  GN  tangent  to  it  at  G\ 
and  these  are  the  extreme  visible  elements  in  the  front  view  of  the 


100 


DESCRIPTIVE    GEOMETRY. 


portion  shown,  wliicli  embraces  three  fourths  of  the  circumference  of 
the  cylinder,  the  rectilinear  elements  terminating,  as  in  Fig.  135, 
at  their  points  of  tangency  to  the  directrix. 

156.  To  assume  a  point  on  the  surface,  assume  the  horizontal 
projection,  asj9.  Through^  draw  the  horizontal  projection  of  an 
element;  it  is  tangent  to  the  circle  of  the  cylinder's  base  at  c,  ver- 
tically projected  at  c'  on  the  helix,  and  cuts  the  involute  at  <?,  ver- 
tically projected  at  e'  in  AB :  j^'  must  lie  on  c'e'  the  vertical  projec- 


tion of  the  element.  By  reversing  this  construction  the  horizontal 
projection  may  be  found  if  ^'  is  assumed. 

157.  Problem  4.  To  draw  a  plane  tangent  to  this  convolute 
and  parallel  to  a  given  right  line. 

The  tangents  to  the  helix  make  equal  angles  with  the  plane  of 
the  base,  and  it  is  apparent  that  this  surface,  though  not  warped, 
has  a  cone  directer;  which,  as  shown  in  Fig.  151,  is  a  cone  of 
revolution,  whose  elements  make  the  same  angle  with  the  plane  of 
the  base.  If  it  be  required  to  draw  a  plane  tangent  to  the  convo- 
lute and  parallel  to  the  given  line  XY\  lirst  draw  a  plane  parallel 
to  this  line  and  tangent  to  the  cone  directer  (151):  ss  is  its  hori- 


DESCKIPTIVE    GEOMETRY.  101 

zoiital  trace,  and  the  horizontal  trace  tT  oi  the  required  plane  is 
parallel  to  ss  and  tangent  to  the  involute.  Next  draw  a  tangent  to 
the  circular  base  of  the  cylinder,  perpendicular  to  tT\  this  is  the  hori- 
zontal projection  of  the  element  of  contact,  and  locates  the  point  of 
tangency  d^  which  is  vertically  projected  at  d'  in  AB.  The  point 
of  contact  ic  with  the  circle  is  projected  to  u  upon  the  helix,  thus 
determining  d'u'  the  vertical  projection  of  the  element  of  contact, 
whose  vertical  trace  is  B\  and  Tr't'  is  the  vertical  trace  of  the 
tangent  plane.  Should  7''  be  inaccessible,  draw  through  any  point 
L  upon  DU  2,  parallel  to  tT\  it  is  a  horizontal  line  of  the  plane, 
and  pierces  V  in  K^  a  point  of  the  required  vertical  trace. 

158.  To  draw  ajplane  tangent  at  any  assumed  pointy  as  Pi 
the  element  CE  through  tliis  point  is  one  line  of  the  plane ;  the 
horizontal  trace  is  drawn  through  ^,  perpendicular  to  ce^  and  the 
vertical  trace  is  then  found  as  above. 

To  dravj  a  tangent  plane  through  a  point  without  the  surface. 

Draw  first  a  horizontal  plane  through  the  given  point  (150): 
its  intersection  with  the  surface  will  be  another  involute,  to  which 
a  tangent  is  to  be  drawn  through  the  given  point.  This  will  be 
one  line  of  the  required  plane ;  the  element  through  the  point  of 
contact,  which  is  found  as  in  (157),  is  another,  and  by  means  of 
these  the  traces  may  be  determined. 

159.  In  these  illustrations,  the  lower  nappe  only  of  the  convolute 
has  been  represented  and  considered,  in  order  to  avoid  confusion 
in  the  diagrams.  But  it  must  not  be  f(>rgotten  that  the  rectilinear 
elements  can  be  indefinitely  extended  both  ways  from  the  points 
of  contact  with  the  helix ;  so  that  a  plane  tangent  to  the  surface  at 
any  point  is  tangent  to  it  all  along  a  line  lying  on  both  nappes : 
and  it  makes  no  diiference  whether  the  upper  or  the  lower  one  is 
em])loyed  in  the  process  of  construction. 

160.  Considering  any  point  as  P,  Fig.  150,  at  a  given  dis- 
tance CP  from  the  point  of  contact  C  between  the  element  and  the 
directrix :  it  is  clear  that  in  the  generation  of  the  surface  the  path 
of  this  point  will  be  a  helix  whose  pitch  is  the  same  as  that  of  the 
directrix  itself.  This  particular  convolute,  therefore,  is  one  of  the 
numerous  family  of  helicoids^  of  which  all  the  others  are  eithei' 
warped  or  double-curved  surfaces.     And  though  it  is  usually  pre- 


102  DESCRIPTIVE    GEOMETRY. 

seiited  and  represented  in  a  manner  so  imperfect  and  ol)scure  as  to 
conceal  the  fact,  it  possesses  a  certain  practical  interest,  because  it  is 
in  fact  the  surface  of  a  screw-thread,  which,  as  will  subsequently 
be  shown,  can  be  cut  in  a  lathe  in  the  usual  manner. 

161.  In  general,  a  plane  cannot  be  passed  through  a  given 
right  line  and  tangent  to  a  single-curved  surface.  The  problem, 
however,  is  possible  in  the  following  cases,  viz. ,  if  the  given  line 
lies  on  the  convex  side  of  a  cylinder,  and  is  parallel  to  its  rectilinear 
elements ;  if  it  passes  through  the  vertex  of  a  cone ;  or  if  it  is  tan- 
gent to  a  line  of  any  single- curved  surface. 

PLANES    TANGENT    TO    DOUBLE-CURVED    SURFACES. 

162.  A  plane  tangent  to  any  double-curved  surface  at  a  given  point 

can  in  general  be  determined  (140)  by  drawing  at  that  point  a  tan- 
gent to  each  of  two  lines  of  the  surface  which  ]3ass  through  that 
point,  the  selection  depending  upon  considerations  of  convenience. 
Of  the  surfaces  of  this  class  which  are  used  in  the  mechanic  arts, 
those  of  revolution  constitute  by  far  tlie  larger  portion,  by  reason  of 
the  facility  with  which  they  can  be  formed  in  the  lathe  or  on  the 
potter's  wheel;  they  serve  as  well  as  any  for  illustrative  purposes, 
and  to  them  we  shall  confine  our  attention.  In  dealing  witli  them 
in  the  manner  stated,  the  two  curves  which  would  naturally  be  se- 
lected are  the  meridian  line  and  the  circumference  of  the  transverse 
section  through  the  given  point. 

163.  If  a  right  cone  be  tangent  to  a  surface  of  revolution 
which  has  the  same  axis,  it  will  be  tangent  all  round  the  circumfer- 
ence of  a  circle.  Any  plane  tangent  to  this  cone  will  be  tangent 
all  along  an  element;  therefore  the  plane  will  be  tangent  to  the 
surface  at  the  point  in  which  this  element  of  contact  cuts  that 
circle  of  contact.  Such  an  auxiliary  cone  may  sometimes  be  used 
to  advantage  in  determining  a  plane  tangent  to  a  double-curved 
surface. 

164.  If  the  contour  of  a  surface  of  revolution  be  such  that  the 
direction  of  the  normal  to  it  can  be  readily  determined,  a  plane  tan- 
gent to  the  surface  at  a  given  point  can .  be  constructed  with  great 
facility,  since  it  is  perpendicular  to  tlie  normal  at  its  extremity. 


DESCRIPTIVE    GEOMETRY. 


ws 


165.  Problem  1.  To  draw  a  plane  tangent  to  a  spliere  at  a 
given  point  on  the  surface. 

Construction.  In  Fig.  152,  let  C  be  the  centre  of  tlie  sphere, 
and  P  the  given  point,  assumed  as  in  Fig.  138;  then  CP  is  the 
radius  of  contact.  The  horizontal  projecting  plane  of  this  radius 
contains  a  horizontal  diameter  of  the  spliere ;  revolving  that  jjlane 
about  this  diameter  until  it  is  parallel  to  H,  P  falls  dXp'\  and  the 


Fig.  153 


horizontal  trace  appears  as  H" H"  parallel  to  cp^  the  distance  cx'^ 
being  equal  to  ex' .  Draw  at^''  a  tangent  to  the  great  circle;  it 
cuts  H" H"  in  d" ^  the  revolved  position  of  a  point  d  in  the  re- 
quired horizontal  trace,  v/hich  is  perpendicular  to  cp  (43).  Draw 
at  P  a  tangent  to  the  horizontal  circle  of  the  sphere  through  that 
point ;  it  is  a  line  of  the  tangent  plane,  and  pierces  V  in  ^ :  the 
line  PD  pierces  V  in  N\  therefore  the  points  r\  n' ^  determine  the 
vertical  trace,  which  must  be  perpendicular  to  c'p'  (43),  and  meet 
the  horizontal  trace  at  T  in  AB. 

I*^OTE.  Regarding  tlie  sphere  as  a  surface  of  revolution  with  a 
vertical  axis,  it  is  seen  that  DPO  is  an  element  ot  a  right  cone 
tangent  to  the  sphere  around  the  circle  described  by  P\  the  base 


104  DESCKIPTIVE    GEOMETRY. 

of  this  cone  in  H  is  a  circle  with  radius  cd  —  x"d" \   and  Tt  is  tan- 
gent to  this  circle. 

166.  Pkoblem  2.    To  draw  a  plane  tangent  to  any  surface  of 
revolution  at  a  given  point  on  the  surface. 

Coiistructioii.  In  Fig.  153,  the  assumed  point  P  lies  upon  the 
circle  whose  vertical  j^rojection  is  m'n'.  Let  c  be  the  centre  of  cur- 
vature of  the  contour  7ig%  then  en'  is  normal  to  the  curve  at  n  \ 
its  prolongation  cuts  the  axis  in  (9,  and  o'n'm'  is  the  vertical  pro- 
jection of  a  cone  normal  to  the  given  surface.  Tlierefore  o'p'  is 
the  vertical  and  op  is  the  horizontal  projection  of  the  normal  to  the 
surface  at  P.  Through  P  di^avf,  as  in  Fig.  67,  a  plane  tTt'  per- 
pendicular to  P(9;   it  is  the  tangent  plane  required. 

167.  Problem  3.  2^o  draw  through  a  given  line  a  j)l(ine  tan- 
gent to  a  given  sphere/ 

First  Method.  Analysis.  Eegarding  the  given  line  as  the  in- 
tersection of  two  planes  tangent  to  the  sphere,  each  plane  is  perpen- 
dicular to  a  radius  at  its  extremity;  therefore  the  plane  of  these 
two  radii  is  perpendicular  to  the  given  line.  If  then  a  plane  be 
passed  through  the  centre  of  the  sphere  and  perpenaicular  to  the 
given  line,  and  from  the  point  in  which  it  cuts  the  line  a  tangent 
be  drawn  to  the  great  circle  cut  from  the  sphere ;  then  the  plane 
determined  bj  that  tangent  and  the  given  line  is  the  one  required. 

Construction.  Let  (7,  Fig.  154,  be  the  centre  of  the  given  sphere, 
MW  the  given  line. 

On  the  horizontal  projecting  plane  of  the  line  make  a  sup- 
plementary projection;  in  this  II'  11'  is  the  horizontal  plane,  c^  the 
centre  of  the  sphere,  m^n^  the  given  line,  and  LL  the  plane  ])ei  - 
pendicular  to  the  line,  of  which  II  is  the  horizontal  trace.  This 
plane  contains  a  horizontal  diameter  of  the  sphere,  of  which  ge  is 
the  horizontal  projection ;  and  it  cuts  MN  in  the  point  A.  Ee- 
volve  the  plane  about  ge  until  it  becomes  horizontal ;  its  trace  II 
then  appears  as  l"l" ^  and  a  falls  at  a" .  Draw  a" d"  tangent  to  the 
great  circle  of  the  sphere,  and  find  o"  the  point  of  contact.  Mak- 
ing the  counter-revolution,  a"  returns  to  ^,  d"  falls  at  d  in  II 
(vertically  projected  at  d'  in  AB),  and  o"  goes  to  o  on  da.,  whence 
it  is  vertically  projected  to  d  on  d'a' . 

The  tangent  line  7>^  .pierces  H  in  Z^,  the  given  line  pierces  it 


DESCRIPTIVE    GEOMETRY. 


105 


in  iV^;  therefore  tlie  points  d  and  n  determine  the  horizontal  trace 
tT^  which  must  also  be  perpendicular  to  co  :  JfiV^  pierces  V  in  B,  and 
r  is  a  point  in  the  vertical  trace  Tf  of  the  required  tangent  plane ; 
and  tliis  trace  must  also  be  perpendicular  to  c'o\  the  vertical  pro- 
jection of  the  radius  of  contact. 

168.  Second  Method.     Analysis.      If  any  point  of  the  line  be 
taken  as  the  vertex  of  a  cone  tangent  to  the  sphere,  a  plane  con- 


1           ^ 

Zyv^ 

,'/ 

X'                         TF          '/     \n' 

^ 

~^^\/               \ja"     ^"^^ 

m^ihg 

taining  the  given  line  and  tangent  to  this  cone  will  also  be  tangent 
to  the  sphere.  The  cone  is  tangent  to  the  sphere  all  round  the 
circumference  of  a  small  circle ;  the  plane  is  tangent  to  the  cone 
all  along  an  element;  and  the  intersection  of  the  circle  and  the 
element  is  the  point  of  contact  between  the  sphere  and  the  plane. 

Construction.     In  Fig.  155,  C' is  the  centre  of  the  sphere,  MN 
the  given  line.      The  point  Z>,  in  which  the  line  is  cut  by  a  liori- 


106 


DESCRIPTIVE    GEOMETRY. 


m/   ;      n\    'X'         z\ 


V 


0^ 


.Tig.  157 


DESCRIPTIVE   GEOMETRY.  WT 

zoTital  plane  LL  through  C,  is  selected  as  the  vertex  of  the  auxili- 
ary cone,  merely  for  convenience ;  its  axis  being  thus  made  hori- 
zontal, the  plane  of  the  circle  of  contact  is  vertical,  and  appears  as 
a  right  line  nw  in  the  horizontal  projection.  This  plane  cuts  MJV 
in  a  point  whose  horizontal  projection  is  e  and  vertical  projection 
e' ;  in  the  supplementary  projection  on  this  plane  it  appears  as  e^ : 
through  e^  draw  a  tangent  to  the  circle  of  contact,  find  the  point  of 
tangency  <?,,  and  produce  it  to  cut  H' H'  in  g^.  These  two  points 
are  projected  back  to  o  and  g\  g  is  one  point,  and  m  is  another,  in 
the  horizontal  trace  of  the  required  plane,  which  must  also  be  per- 
pendicular to  CO.  The  vertical  projection  o'  will  be  in  a  perpen- 
dicular to  AB  tJirough  o,  at  a  distance  o'z'  above  AB  equal  to  o,s,  in 
the  supplementary  projection ;  the  given  line  pierces  T  in  iV^,  and 
the  element  OD  of  the  cone  pierces  it  in  S;  therefore  Ts'ii't'  is 
the  vertical  trace  of  the  tangent  plane,  which  nmst  be  perpendicular 
to  c'o\  the  vertical  projection  of  the  radius  of  contact.  In  order  to 
avoid  confusion,  the  vertical  projection  of  the  auxiliary  cone  is  not 
drawn,  no  use  being  made  of  it  in  the  construction. 

169.  Third  Method.  Analysis.  If  any  two  points  of  the  given 
line  be  taken  as  the  vertices  of  cones  tangent  to  the  sphere,  the 
planes  of  the  circles  of  contact  will  intersect  in  a  common  chord, 
whose  extremities  w^ill  lie  on  all  three  surfaces.  A  plane  contain- 
ing the  given  line  and  either  of  these  points  will  be  tangent  to  the 
sphere. 

Construction.  This  is  illustrated  pictorially  in  Fig.  156,  where 
S  is  the  sphere,  MN  the  given  line,  and  7>,  E.,  are  the  vertices  of 
the  two  cones.  The  construction  in  projection  is  given  in  Fig.  157, 
(7  being  the  centre  of  the  sphere,  and  vJ/iTthe  given  line.  A  hori- 
zontal plane  LL  through  C  cuts  the  line  in  D^  and  a  plane  FF 
through  C  and  parallel  to  V  cutfS  it  in  E.  Take  D  and  E  as  the 
vertices  of  the  two  cones ;  then  the  axis  of  the  first  being  horizon- 
tal, the  plane  of  the  circle  of  contact  will  be  vertical ;  kg  is  its 
horizontal  trace.  The  axis  of  the  second  will  be  parallel,  and  the 
plane  of  the  circle  of  contact  perpendicular,  to  the  vertical  plane ; 
Ic'g'  is  the  vertical  trace.  Therefore  Ug^  lc'g\  are  the  projections  of 
the  intersection  of  these  two  planes.  This  line  pierces  H  in  6^; 
in  order  to  find  where  it  pierces  the  surface  of  the  sphere,  make  a 


108 


DESCRIPTIVE    GEOMETRY. 


supplementary  projection  on  a  plane  perpendicular  to  the  axis  of 
the  first  cone.  In  this  projection  the  line  of  contact  is  seen  in  its 
true  form  as  the  circle  of  wliich  ^i^  is  the  centre,  and  the  line  k^g^ 
cuts  its  circumference  at  6>, ,  Avhich  is  projected  back  to  o,  and 
thence  vertically  to  o\  the  distance  o'b'  being  equal  to  o^z^.  Then 
DO  is>  Si  line  of  the  required  plane;  its  vertical  and  horizontal 
traces  are  xS'and  I^;  those  of  the  given  line  are  JV  and  J/;  there- 
fore mr  7^  is  the  horizontal  and  Tn's'  is  the  vertical  trace  of  the 
tangent  plane  :  and  these  traces  are  respectively  perpendicular  to  the 
projections  of  CO  the  radius  of  contact. 

The  vertical  projection  of  the  first  cone  and  the  horizontal  pro- 
jection of  the  second  are  omitted ;  no  use  being  made  of  either  in 
effecting  the  solution,  their  introduction  would  simply  confuse  the 
diagram.  A  like  result  would  have  followed  had  the  determina- 
tion of  the  other  tangent  plane  been  represented,  and  the  construc- 
tion of  one  is  sufficient  to  illustrate  the  principles  of  either  method. 

170.  Problem  4.  To  find  the  jpoint  of  contact  of  any  surf  ace 
of  revolution  with  a  plane  perjpendicular  to  a  given  line. 

Construction.     In  Fig.  158,  the  axis  of  the  surface  is  vertical. 


No  ground  line  is  drawn,  but  the  axis  may  be  regarded  as  lying  in 
a  vertical  plane,  represented  in  the  top  view  by  the  horizontal  cen 


DESCRIPTIVE   GEOMETRY. 


109 


tre  line.  Revolving  the  given  line  MN  about  a  vertical  axis  until 
it  lies  in  this  plane,  it  takes  the  position  MN^.  The  plane  LL  is 
drawn  tangent  to  the  contour  and  perpendicular  to  m'n^^  and  0\  is 
the  revolved  position  of  the  point  of  contact.  In  the  counter- 
revolution this  point  describes  an  arc  of  a  horizontal  circle,  the  angle 
o^co  being  equal  to  the  angle  n^mn^  and  0  is  the  true  position  of 
the  required  point. 

In  Fig.  159,  the  axis  is  horizontal,  and  a  vertical  plane  contain- 
ing it  is  represented  by  the  vertical  centre  line  in  the  end  view. 
The  given  line  MN  and  the  surface  together  are  revolved  about 
the  axis  until  the  line  comes  into  that  vertical  plane ;  and  the  re- 
maining steps  require  no  explanation. 

In  Fig.  160,  the  axis  lies  in  the  vertical  plane,  but  is  inclined, 
the  object  accordingly  appearing  foreshortened  in  the  top  view. 
The  line  to  which  the  tangent  plane  is  to  be  perpendicular  is  given 


by  the  projections  mn^  m'n' ;  in  the  supplementary  end  view  it  ap- 
pears as  m,i^j,  the  distance  n^x^  being  equal  to  nx  in  the  top  view. 
This  line  is  revolved  about  the  axis  of  the  surface  into  the  plane 
FF  as  in  Fig.  159,  n^  going  to  n^,  which  is  projected  back  to  n"  \ 


110 


DESCRIPTIVE    GEOMETRY. 


the  plane  LL  is  then  drawn  tangent  to  the  contour,  and  the  point 
of  contact  o"  is  projected  to  o^,  revolved  to  <?,,  and  re-projected  to 
o' .  In  the  top  view,  o  is  directly  under  o\  at  a  distance  from  the 
centre  line  equal  to  the  distance  of  o^  from  FF  in  the  end  view. 

KoTE.  The  determination  of  the  outline  in  the  top  view,  Fig. 
160,  depends  upon  this  principle,  viz.,  that  the  yisible  contour  of 
any  object  is  the  envelope  of  all  the  lines  upon  its  surface.  In  the 
present  case,  all  the  transverse  sections  are  circles,  whose  horizontal 
projections  are  ellipses,  and  the  contour  is  tangent  to  all  these 
ellipses, 

171.  Problem  5.  To  draw  a  plane  making  given  angles  with 
the  pi'incipal  planes  of  projection. 

Argument.  In  the  profile.  Fig.  161,  (9  and  P  are  the  vertices 
of  two  cones  tangent  to  the  same  sphere,  the  axis  of  one  being  ver- 


\ 

o 

i 

P> 

\   \ 

M 

Fig.  161 


tical,  that  of  the  other  perpendicular  to  V.  All  planes  tangent  to 
the  first  are  equally  inclined  to  H ;  all  those  tangent  to  the  second 
are  equally  inclined  to  Y :  a  plane  through  the  line  MN  joining 
their  vertices,  if  tangent  to  one,  will  be  tangent  to  both,  as  in  Fig. 
157.  J/ will  be  a  point  in  the  horizontal  trace,  N  2,  point  in  the 
vertical  trace ;  and  each  trace  will  be  tangent  to  the  base  of  the 
cone  which  lies  in  the  correspondhig  plane  of  projection. 

In  the  application,  the  angles  at  the  bases  of  the  cones  must  be 
made  equal  to  the  assigned  angles  which  the  required  plane  is  to 
make  with  H  and  V  respectively. 

Construction.     In  Fig.  162,  the  profile  is  first  drawn,  the  cen- 


DESCRIPTIVE    GEOMETRY.  Ill 

tre  c^  of  the  sphere  being  for  convenience  placed  in  the  ground 
line.  Tangent  to  the  outline  of  the  sphere,  draw  x^s^^  making  the 
angle  x,s^g^  equal  to  the  assigned  angle  with  H ;  and  also  z{r^ ,  the 
angle  z{t\G^  being  equal  to  the  assigned  angle  with  V.  Then  draw 
an  indefinite  perpendicular  to  AB  through  any  convenient  point  (7, 
and  on  it  set  off  Cx  above,  equal  to  (?,a?, ;  also  Cz  below,  equal  to 
c,2^.  About  C  describe  above  AB  an  arc  with  radius  Cr'  =  c^r^, 
and  draw  x'T  tangent  to  it;  also  about  C  describe  below  AB  an  arc 
with  radius  Os  =  c^s^,  and  draw  sT  tangent  to  it.  These  two  tan- 
gents  will  be  the  traces  of  the  required  plane,  which  must  intersect 
at  T  in  the  ground  line. 

JN'oTE.  A  different  solution  of  this  problem  has  already  been 
given  in  (94),  but  the  one  here  presented  as  a  neat  application  of 
tangent  planes  is  preferable,  being  more  reliable  as  well  as  more 
expeditious. 


li;;i  DESCRIPTIVE   GEOMETRY. 


CHAPTEE  Y. 

OF    INTERSECTIONS    AND    DEVELOPMENTS. 

Intersection  of  Surfaces  by  Planes.  Development  of  Single-curved  Surfaces. 
Tangents  to  Curves  of  Intersection,  before  and  after  Development.  Prob- 
lem of  the  Shortest  Path.  Intersections  of  Single-curved  Surfaces.  In- 
finite Branches.  Intersections  of  Double-curved  Surfaces.  Intersection 
of  a  Cone  with  a  Sphere.     Development  of  the  Oblique  Cone. 

172.  If  a  line  be  drawn  upon  one  surface,  tlie  point  in  wliicli  it 
pierces  any  other  surface  will  lie  upon  the  intersection  of  the  two. 
In  order  to  find  that  point,  an  auxiliary  surface  is  passed  through 
the  line ;  this  intersects  the  second  surface  in  another  line,  which  in 
turn  cuts  the  first  line  in  the  point  sought. 

This  is  illustrated  in  finding  the  point  in  which  a  given  right 
line  pierces  a  plane ;  the  auxiliary  plane  intersects  the  given  plane 
in  another  right  line,  and  that  cuts  the  first  one  in  the  required 
point.  It  is  clear  that  the  line  here  spoken  of  as  given  might  have 
been  cut  from  any  given  rnled  surface  by  the  auxiliary  23lane ;  then 
the  point,  located  as  above,  would  have  been  one  point  in  the 
intersection  of  that  surface  by  the  given  plane ;  and  other  points  could 
be  found  by  means  of  other  auxiliary  planes. 

Again,  were  any  other  surface  substituted  for  the  given  plane, 
the  construction  would  be  modified  only  in  this,  that  the  auxiliary 
plane  might  intersect  it  in  a  curve  instead  of  in  a  right  line :  but 
this  curve  would  still  cut  the  rectilinear  element  of  the  first  surface 
in  a  point  of  the  line  of  intersection  of  the  two  surfaces. 

173.  The  point  last  mentioned  is  the  one  in  which  the  first 
surface  is  pierced  by  the  curve  cut  from  the  second ;  and  evidently 
would  be,  whether  the  line  cut  from  the  first  by  the  auxiliary  plane 
were  straight  or  otherwise.  Which  shows  that  the  point  in  which 
any  plane  curve  pierces  a  given  surface  may  be  found  by  first  de- 


DESCRIPTIVE    GEOMETRY.  113 

termining  the  intersection  of  its  plane  with  that  surface ;  it  will  be 
a  line  cutting  the  given  curve  in  the  required  point. 

But  even  in  the  case  of  a  right  line  piercing  a  plane  it  is  not 
necessary  that  the  auxiliary  surface  should  be  a  plane :  if,  for  ex- 
ample, a  cone  or  a  cylinder  be  passed  through  the  line,  intersecting 
the  given  plane  in  a  circle,  it  will  now  be  apparent  that  the  circum- 
ference will  cut  the  given  line  in  the  point  required.  And  in  deal- 
ing with  a  line  of  double  curvature  the  simplest  auxiliary  surface 
that  can  be  used  is  cylindrical.  Thus,  in  order  to  find  where  the 
curve  DEG^  Fig.  131,  pierces  the  vertical  plane,  we  make  use  of 
the  horizontal  projecting  cylinder,  from  which  V  cuts  the  element 
M',  and  this  intersects  the  curve  in  h'^  the  point  sought. 

The  same  figure  illustrates  the  process  of  finding  the  intersection 
of  two  cylindrical  surfaces ;  one,  whose  base  is  ceh^  having  vertical 
elements,  while  the  elements  of  the  other,  whose  base  is  c'e'k' ^  are 
perpendicular  to  Y.  An  auxiliary  plane  ee\  parallel  to  the  rec- 
tilinear elements  of  both  cylinders,  cuts  from  each  a  riglit  line,  and 
these  intersect  in  E\  and  by  other  planes  parallel  to  this  the  points 
D  and  G  are  determined ;  these  points  lie  upon  both  surfaces,  and 
the  curve  cEk'  is  the  required  intersection. 

174.  From  the  preceding  it  will  be  perceived  that  the  problems 
of  finding  the  points  in  which  surfaces  are  pierced  by  lines,  and  the 
lines  in  which  surfaces  intersect  each  other,  are  correlated  and  inter- 
woven, involve  the  same  principles,  and  require  for  their  solution  a 
previous  knowledge  of  the  intersections  of  the  surfaces  under  con- 
sideration by  certain  others,  which  may  be  used  as  auxiliaries.  In 
all  of  them  the  ultimate  determinations  consist  in  the  location  of 
points  by  the  intersections  of  lines;  and  the  auxiliary  surfaces 
should  be  so  selected  that  these  lines  shall  cut  each  other  as  nearly 
as  possible  at  right  angles. 

175.  Tangents  to  Curves  of  Intersection.  If  a  surface  is  cut  by 
a  plane,  the  tangent  to  the  line  of  intersection  at  any  point  will  lie 
in  that  plane,  and  also  in  the  plane  tangent  to  the  surface  at  the 
given  point ;  therefore  it  will  be  the  intersection  of  those  planes. 

If  two  surfaces  intersect,  the  tangent  to  the  common  line  at  any 
point  will  be  the  intersection  of  two  planes,  one  tangent  to  each  sur- 
face at  that  point. 


114  DESCRIPTIVE   GEOMETRY. 

Note.  If,  as  sometimes  happens,  the  two  planes  whose  inter- 
section should  determine  the  tangent  coincide,  this  method  obvi- 
ously fails  to  give  any  result.  In  this  event,  the  direction  of  the 
tangent  can  be  determined  only  by  methods  depending  upon  the 
mathematical  properties  of  the  curve,  of  which  this  branch  of  sci- 
ence does  not  treat. 

176.  Development  of  Surfaces.  A  prism  is  capable  of  rolling,  in 
a  hobbling  and  imperfect  manner,  upon  a  plane;  turning  about 
each  edge  in  succession  as  an  axis,  so  that  one  face  after  another  is 
brouglit  into  coincidence  with  the  plane.  If  the  number  of  edges 
be  increased,  the  hobbling  will  diminish,  and  when  the  number  be- 
comes infinite,  it  will  disappear  entirely  :  the  rolling  is  perfect,  the 
change  from  one  axis  to  another  going  on  continuously,  since  the 
edges  are  now  consecutive  elements  of  a  cylinder.  In  this  way  all 
the  elements  of  the  cylinder,  without  change  of  relative  position, 
can  be  brought  into  the  plane,  the  area  rolled  over  being  equal  to 
that  of  the  surface  wdiich  has  rolled  over  it. 

A  pyramid,  treated  in  like  manner,  ultimately  becomes  a  cone, 
which  possesses  the  same  property :  indeed  it  is  a  sufiiciently 
famihar  fact  that  either  of  these  surfaces  can  be  unrolled  into  a 
plane,  without  extension,  compression,  or  distortion  of  any  kind. 

177.  This  process  is  called  the  development  of  the  surface. 
And  the  two  illustrations  above  given  show  clearly  upon  what  its 
possibility  depends :  the  plane  of  development  is  not  only  tangent 
to  the  surface,  but  contains  two  of  its  consecutive  elements,  and 
therefore  the  elementary  surface  included  between  them.  These 
elements  must  be  rectilinear,  since  each  in  turn  is  an  axis  of  rota- 
tion, and  an  axis  is  a  right  line ;  no  two  consecutive  elements  of  a 
warped  surface  lie  in  the  same  plane,  therefore  all  single-curved 
surfaces,  and  no  others,  are  capable  of  development. 

It  is  evident  that  if  any  surface  can  roll  upon  a  plane,  the  plane 
is  equally  capable  of  rolling  upon  the  surface ;  and  this  develop- 
Mient  by  rolling  on  a  fixed  plane  is,  in  a  limited  sense,  the  converse 
of  the  generation  of  the  surface  by  a  moving  plane. 

But  in  order  to  execute  this  process,  it  is  not  enough  to  know 
that  the  surface  is  developable :  it  is  necessary  also  to  know  before- 
hand the  developed  forms  of  one  or  more  lines  of  the  surface  which 


DESCRIPTIVE   GEOMETRY. 


no 


intersect  the  rectilinear  elements,  in  order  to  determine  tlie  relative 
positions  of  these  elements  on  the  plane  of  development. 


INTERSECTIONS    OF    SURFACES    BY    PLANES. 

178.  Problem  1.  To  find  the  intersection  of  a  right  circular 
cylirider  with  a  jplane. 

Construction.  In  Fig.  163,  the  axis  of  the  cylinder  is  vertical, 
its  base  lies  in  the  horizontal  plane,  and  the  given  plane  tTt'  is  per- 
pendicular to  V. 

The  points  in  which  the  elements  through  the  points  cc,  1,  2, 
etc.,  of  the  base  pierce  the  plane  are  seen  directly  at  u\  1',  2'. 


etc. ,  in  the  front  view ;  and  u'^'z'  is  the  vertical,  and  xZy  is  the 
horizontal,  projection  of  the  required  intersection. 

To  draw  the  tangent  at  any  point  P.  This  must  lie  in  the  tangent 
plane,  which  is  vertical,  and  has  the  horizontal  trace  po  tangent  to 
the  circumference  of  the  base ;  the  tangent  also  lies  in  the  cutting 
plane,  therefore  its  vertical  projection  ^'o'  coincides  with  Tt'. 


116  DESCRIPTIVE    GEOMETRY. 

To  show  the  true  form  of  the  curye.  Make  the  supplementary 
projection  S,  looking  perpendicularly  against  the  given  plane.  In 
this  view,  tt  will  aj)pear  as  tf^  perpendicular  to  Tt\  u'z'  will  be 
seen  in  its  tnie  length  as  u^z^  parallel  to  Tt' ^  and  tlie  chords  verti- 
cally projected  at  V ^  2\  etc.,  will  also  be  seen  in  their  true  lengths 
as  1,1,,  2,2,,  etc. ;  these  lengths  are  equal  to  those  of  11,  22,  etc.,  in 
tlie  top  view ;  this  curve  is  an  ellipse  whose  major  axis  is  u'z,  and 
whose  minor  axis  is  equal  to  33,  the  diameter  of  the  given  cylinder. 

To  draw  the  tangent  to  this  curve  in  its  own  plane.  The  point 
"whose  horizontal  projection  is  p  and  vertical  projection  p'  is  found 
in  the  supplementary  pro jectibn  at^, :  the  tangent  at  this  point 
was  seen  in  the  top  view  to  cut  the  horizontal  trace  at  o,  and  x^ 
produced  cuts  that  trace  in  r.  In  the  supplementary  j)rojection 
2^Ui  produced  cuts  the  horizontal  trace  in  /•, ;  now  set  off  r^o^  =  7'o, 
and  draw^,6>, ;  it  will  be  the  required  tangent  at  the  point  j9,. 

179.  To  develop  the  lower  part  of  the  cylinder.  Suppose  the 
cylinder,  formed  of  thin  sheet  metal,  to  be  cut  through  the  element 
os'u\  and  unrolled.  The  vertical  elements  will  remain  vertical,  and 
since  the  base  is  a  continuous  curve  perpendicular  to  them  all,  it  will 
develop  into  a  right  line  x^x^^  as  shown  in  D,  whose  length  is  equal 
to  the  circumference.  If  in  the  top  view  this  circumference  is 
divided  into  equal  parts  at  1,  2,  etc.,  then  the  developed  base  will 
be  similarly  divided  at  1^,  2,,  etc.  ;  at  each  point  of  subdivision 
there  will  be  a  vertical  ordinate,  representing  an  element;  and 
since  the  lengths  of  these  elements  remain  unchanged,  we  sliall  have 
I3I,  =  1^1',  2323  =  2'2',  etc.,  and  u^S^^u^  will  be  the  development 
of  the  curve  of  intersection. 

To  draw  the  tangent  to  the  developed  curve.  The  tangent  to  the 
intersection  at  P  contains  two  consecutive  points  of  the  curve,  and 
will  contain  them  after  development,  and  will  therefore  be  tangent 
to  the  developed  curve.  When  the  plane  of  development  becomes 
tangent  to  the  cylinder  at  P,  it  will  contain  the  element  PN,  the 
tangent  PO,  and  the  subtangent  iV6^ ;  and  these  will  remain  un- 
changed in  magnitude  and  in  relative  position.  In  the  develop- 
ment D  this  element  falls  at  p^n^  *,  therefore,  setting  off  n^o^  =  no, 
we  have  p./),  as  the  tangent  to  the  developed  curve. 

In  drawing  the  tangents  at  u^  and  2^  in  this  manner,  it  is 


DESCRIPTIVE   GEOMETRY.  117 

obvious  tliat  tlie  subtangents  will  be  infinite ;   the  tangents  at  those 
j^oints  are  therefore  parallel  to  x^x^. 

180.  The  Problem  of  the  Shortest  Path.  Let  it  be  required  to 
find  the  shortest  path  on  the  surface  of  the  cylinder,  between  the 
points  x^  x\  and  y,  k' .  In  the  development  these  points  fall  re- 
spectively at  a?2,  A^2,  and  the  least  distance  between  them  is  the  right 
line  which  joins  them.  This  line  cuts  the  various  elements  at  points 
wliose  distance  from  the  base  wall  remain  the  same  when  the  de- 
veloped sheet  is  re-formed  into  a  cylinder.  These  points,  there- 
fore, are  projected  back  to  tlie  original  positions  of  the  elements, 
as  h^  to  h\  g^  to  g\  x^  to  a?',  etc.,  thus  forming  in  the  vertical  pro- 
jection  the  curve  x'g'k'^  which  represents  the  required  shortest 
path.  In  the  development  D  it  is  seen  that  the  distances  of  the 
points  of  this  curve  from  the  base  of  the  cylinder,  measured  on 
equidistant  elements,  increase  at  a  uniform  rate;  therefore  the 
curve  itself  is  a  helix,  of  which  y'k'  is  half  the  pitch.  In  the  ver- 
tical projection  the  outer  elements  of  the  cylinder  are  tangent  to 
tlie  curve  at  x  and  h'  \  g'  is  a  point  of  contrary  flexure,  and  the 
tangent  at  that  point  is  parallel  to  xji^. 

It  is  also  to  be  noted  that  the  development  of  the  curve  of  inter- 
section is  identical  with  the  projection  of  a  helix  wliose  half-pitch 
is  equal  to  x^y.^ ,  lying  on  a  cylinder  of  w^iicli  the  diameter  is  ti}^ ; 
the  tangent  at  S,  is  parallel  to  u'z' . 

181.  Practical  Applications.  Since  the  ellipse  is  perfectly  sym- 
metrical about  both  axes,  it  may  be  turned  end  for  end  ;  thus  the  two 
portions  of  a  cylinder  cut  by  a  plane  making  an  angle  of  45°  with 
the  axis  may  be  joined  together  as  shown  at  E,  making  what  is 
known  as  a  "square  elbow."  By  using  other  angles,  the  pieces 
may  be  put  together  at  different  inclinations,  as  shown  at  F,  which 
represents  a  ' '  three-section  elbow. ' '  In  order  to  lay  out  the 
sheet  for  the  middle  piece,  cut  it  by  a  plane  "tnni  perpendicular 
to  the  axis;  this  section  wall  develop  into  a  right  line  as  in  D, 
and  the  ordinates  are  set  off  each  way  from  this  to  determine  the 
contour. 

These  problems  of  development  are  of  direct  use  to  w^orkers  in 
sheet  metal.  In  theory  it  makes  no  difference  along  what  element 
the  surface  is  cut ;   but  in  practice,  the  rule  dictated  by  plain  com- 


118 


DESCRIPTIVE   GEOMETRY. 


mon  sense  is  to  cut  it  so  as  to  make  the  shortest  seain^  unless  there 
is  some  good  reason  for  doing  otherwise. 

182.  Problem  2.  To  find  the  intersection  of  an  oblique  cylinder 
hy  an  oblique  jplane. 

Construction.  In  Fig.  164,  the  plane  and  the  cylinder  are  each 


Fig.  164 


inclined  to  both  H  and  T.  Make  a  supplementary  projection  S, 
looking  in  the  direction  tT\  the  horizontal  plane  appears  as  IP  11^ 
perpendicular  to  tT^  and  the  given  plane  as  Tj}^ ,  both  planes  being 
seen  edgewise  and  at  their  true  inclination  to  each  other.  The 
centre  0  of  the  base  is  here  projected  at  o, ,  and  any  point  P  of 
the  axis  at  j?j,  the  altitude  ^^a?,  being  equal  to  j)'x  '^  then  the  new 
projections  of  the  elements  are  parallel  to  o,p^.  The  extreme  visi- 
ble elements  are  determined  by  drawing  at  k  and  y  in  the  horizontal 
projection,  tangents  to  the  base,  perpendicular  to  II' PP' ;  drawing, 
through  k^  and  y^ ,  parallels  to  o^p, ,  these  hnes  are  cut  by  T^t^  at 


DESCRIPTIVE    GEOMETRY.  119 

^1  and  ^, ,  which  obviously  are  the  highest  and  lowest  points  of 
the  curve :  e^  is  projected  back  to  e  on  the  horizontal  projection 
of  the  element  through  K^  and  thence  upward  to  the  vertical  pro- 
jection of  the  same  element,  e'  being  as  far  from  AB  as  e^  is  from 
H' II' .  The  positions  of  z  and  z  are  determined  in  the  same  man- 
ner, and  bj  repeating  the  process  a  point  may  be  found  on  any  ele- 
ment at  pleasure  :  it  is  particularly  desirable  to  locate  with  accuracy 
those  jDoints  which  lie  upon  the  limiting  elements,  as  f  in  the  hori- 
zontal and  c  in  the  vertical  projection,  since  they  are  points  of 
tangency. 

183.  The  above  operation  might  be  defined  as  consisting  i-n  the 
use  of  a  system  of  auxiliary  planes,  parallel  to  the  cylinder  and  to  the 
horizontal  trace  of  the  given  plane ;  these  cut  horizontal  lines  from 
the  plane  and  elements  from  the  cylinder,  whose  intersections  are 
points  in  the  required  curve.  Thus  in  the  supplementary  projec- 
tion S,  dj^^  may  be  regarded  as  representing  a  plane  perpendicular 
to  the  paper ;  its  horizontal  trace  is  (^m,  and  its  intersection  wnth 
tTt'  is  a  line  horizontally  projected  as/"/,  piercing  T  in  iV,  and  ver- 
tically projected  in  n'f  parallel  to  AB.  The  horizontal  trace  cuts 
the  base  in  d  and  m,  and  the  horizontal  projections  of  the  elements 
through  these  points  determine/*  and  Z,  vertically  projected  2Xf'  and 
V ;   and  these  are  points  in  the  required  curve. 

184.  To  draw  a  tangent  to  the  curye  of  intersection.  Let  L  be 
the  point  at  ^vhich  the  tangent  is  to  be  drawn.  The  plane  tangent 
to  the  cylinder  at  this  point  will  contain  the  element  whose  hori- 
zontal projection  is  lni\  and  its  horizontal  trace,  tangent  to  the 
base  at  m,  cuts  Tt  in  r^  vertically  projected  at  r  in  AB :  therefore 
RL  is  the  required  tangent,  which  if  produced  must  pierce  T  in  a 
point  of  the  vertical  trace  of  the  given  plaiie.  This  point  may  be 
determined  by  producing  rl  to  cut  AB  in  5,  which  is  its  horizontal 
projection,  and  s'  in  Tt'  is  its  vertical  projection. 

To  show  the  curve  and  its  tangent  in  their  own  plane.  Make  a 
second  supplementary  projection  S',  looking  perpendicularly  towards 
Tj^, ;  the  horizontal  trace  will  appear  as  t^^  perpendicular  to  T;t^ , 
and  the  different  points  of  the  curve  will  lie  upon  projecting  lines 
drawn  through  <?, ,  ^, ,  etc. ,  parallel  to  t^^.  In  order  to  determine 
their  relative  positions,  draw  in  the  horizontal  projection  any  line 


120  DESCRIPTIVE    GEOMETRY. 

of  the  plane  perpendicular  to  Tt^  as  cm ;  this  will  be  seen  in  S'  as 
a^u^  perpendicular  to  tf^ ,  and  the  distances  of  the  points  of  the 
curve  from  this  line  will  be  the  same  as  their  distances  from  fn^  in 
the  horizontal  projection :  thus,  a^e^  =  ae^  h^2^  —  hz^  "^Ji  =  "^^'f^ 
etc. 

To  draw  the  tangent  at  l^ :  set  off  on  tji^  the  distance  ti^r^  —  iii\ 
and  draw  rj,^ ;   it  is  the  tangent  required. 

185.  From  (172)  it  is  apparent  that  the  vertical  and  horizontal 
projections  miglit  have  been  constructed  by  means  of  the  auxiliary 
planes,  without  using  the  supplementary  projection  S ;  and  indeed 
any  other  system  of  planes  might  have  been  substituted  for  the  one 
here  employed.  In  either  case,  a  rigid  test  of  the  accuracy  of  the 
construction  would  be  to  revolve  the  plane  and  cylinder  together 
around  a  vertical  axis  until,  as  in  Fig.  165,  the  horizontal  trace 
becomes  perpendicular  to  AB ;  the  vertical  projections  of  all  points 
in  the  curve  should  then  fall  in  one  straight  line,  Tt' .  By  the  use 
of  the  projection  S  this  test  is  applied  at  the  outset ;  the  construc- 
tion of  the  projections  on  H  and  T  is  facilitated ;  and  the  true  form 
of  the  curve  is  found  by  making  the  projection  S',  far  more  readily 
than  it  can  be  in  any  other  way. 

186.  Ill  order  to  develop  the  cylinder,  cut  it  by  a  plane  per- 
pendicular to  the  rectilinear  elements ;  this  right  section  will  de- 
velop into  a  right  line.  On  this  line  set  off  the  rectified  distances 
between  the  elements,  and  perpendicular  to  it  draw  the  elements 
themselves;  on  these  perpendiculars  lay  off  the  true  distances,  as 
measured  on  the  elements,  from  the  right  section  to  any  line  of  the 
surface  whose  developed  form  may  be  required. 

To  draw  a  tangent  to  such  a  developed  curve.  Draw  first  a  tan- 
gent to  the  curve  in  its  original  position,  measure  the  angle  be- 
tween it  and  the  element  passing  through  the  point  of  contact,  and 
draw  a  line  making  the  same  angle  with  that  element  at  the  cor- 
responding point  on  the  developed  sheet. 

187.  Problem  3.  To  find  the  intersection  of  an  ohlique  cone 
hy  an  ohlique  plane. 

Construction.  This  differs  from  that  above  explained,  merely 
in  the  respect  that  the  elements  are  convergent  instead  of  parallsl. 
The  point  in  which  any  element  pierces  the  plane,  is  seen  directly 


DESCRIPTIVE    GEOMETRY. 


121 


in  tlie  vertical  projection,  Fig.  166,  since  the  plane  itself  is  per- 
pendicular to  V ;  and  the  horizontal  projection  must  of  course  lie 
on  the  horizontal  projection  of  the  element.     The  true  form  of  the 


curve  is  found  by  constructing  the  supplementary  projection  S', 
precisely  as  in  Fig.  164;  and  the  diagrams  being  lettered  as  nearly 
as  may  be  to  correspond,  no  detailed  explanation  is  necessary. 

The  determination  of  the  tangent  at  Z,  in  the  horizontal  as  well 
as  in  tlie  suppleuientary  projection,  is  also  made  in  the  same  man- 
ner as  in  the  case  of  the  cylinder. 

The  development  of  such  a  cone  cannot  now  be  explained ;  be- 
cause no  method  has  as  yet  been  described  of  drawing  a  line  upon 
the  surface  which  will  develop  into  a  curve  of  known  form  :  this 
matter  will  be  discussed  subsequently. 

188.  Problem  4.  To  find  ilie  intersection  of  a  cone  of  revolu- 
tion hy  a  plane. 

Construction.  In  Fig.  167,  the  axis  of  the  cone  being  vertical 
and  tTt'  perpendicular  to  V,  the  vertical  projections  u\  e\  z\  etc.. 


122 


DESCRIPTIVE   GEOMETRY. 


of  the  points  in  whicli  the  plane  cuts  the  rectilinear  elements  are 
seen  by  inspection  of  the  front  view,  and  may  be  thence  projected 
to  v^  e,  2,  etc. ,  in  the  top  view.  But  in  this  top  view  the  project- 
ing lines  would  cut  the  elements  very  acutely  in  the  neighborhood 
of  J^jP,  thus  making  the  determinations  unreliable.  In  that  region 
another  process  is  preferable  :  the  element  JF^I^  for  instance  pierces 
tTf  at  a  point  whose  vertical  projection  is  ^' ;  a  horizontal  plane 
through  that  point  will  cut  the  cone  in  a  circle  whose  radius  is  m', 
and  its  intersection  with  the  given  plane  will  be  a  right  line  per- 


p' 

\ 

/(\  / 

1 

*'          /  \ 

\ 

'A.  /' 

B 

N^7     ' 

\ 

^ 

=y^'^ 

eT^^'  1 

/    \ 

"'/ 1\ 

<^ 

'  \  / 

\ 

Til  I      1 

/ 

a 

\ 

/ 

k'  b-  d' 

/' 

9' 

T 

/       ^Ti 

=j£- 

\  \ 

ly  z^^p 

r 

r 

d 

B 

^ 

0 

V 

t 

T 

Fig.  167 


d,    a 


pendicular  to  V,  seen  in  its  true  length  in  the  top  view  as  a  chord 
nn  in  the  circle  cut  from  the  cone ;  and  this  process  may  be  re- 
peated to  determine  other  points. 

189.  Since  the  given  plane  in  this  case  cuts  all  the  elements, 
the  intersection  is  an  ellipse,  the  true  length  of  whose  major  axis  is 
the  horizontal  projection  is  also  an  ellipse,  whose  major  axis 
In  order  to  determine  the  minor  axis,  bisect  u'z'  at  c', 


u'z'\ 


IS    uz. 


through  which  point  draw  a  horizontal  plane  hj ;  this  cuts  the  cone 


DESCRIPTIVE   GEOMETRY.  123 

in  a  circle  whose  radius  is  Ih,  and  c'  is  horizontally  projected  as  a 
chord  CG  of  the  circle  described  about  j?  with  this  radius.  It  is 
perfectly  legitimate,  and  practically  it  is  preferable,  thus  to  de- 
termine the  axes,  and  to  construct  the  curve  by  any  of  the  well- 
known  methods  of  drawing  the  ellipse,  not  only  in  the  horizontal 
projection,  but  in  the  supplementary  projection  S,  w^here  the  inter- 
section is  seen  in  its  true  size. 

The  same  modes  of  finding  points  in  the  curve  may  be  used 
when  the  plane  cuts  a  parabola  or  an  hyperbola  from  the  cone ; 
the  use  of  auxiliary  transverse  sections  being  more  particularly  ap- 
plicable wlien  the  angle  at  the  vertex  is  acute. 

190.  To  draw  a  tangent  at  any  point  of  the  intersection,  as  N". 
The  element  through  this  point  pierces  H  at  F^  and/J?,  tangent  to 
the  circle  of  the  base,  is  the  horizontal  trace  of  the  tangent  plane ; 
tliis  cuts  Tt  in  c»,  and  on  is  tlie  horizontal  projection  of  the  tangent. 
A  plane  through  the  axis,  parallel  to  V,  cuts  Tt  in  r\  in  the  sup- 
plementary projection  S,  set  off  r^o^  =  ro,  and  o,n^  will  be  the  tan- 
gent to  tlie  curve  in  its  own  plane,  as  in  Fig.  163. 

If  the  given  plane  be  parallel  to  the  plane  of  any  two  elements, 
tlie  section  will  be  an  hyperbola ;  the  plane  tangent  to  the  cone 
along  either  of  these  two  elements,  will  intersect  the  cutting  plane 
in  a  line  parallel  to  the  element  itself,  and  this  line  will  be  an 
asymptote  to  the  curve.  If  the  cutting  plane  be  parallel  to  one 
element  onlj^,  it  will  be  parallel  to  the  plane  tangent  along  that  ele- 
ment, and  therefore  wdll  not  intersect  it  at  all ;  w^hich  is  as  it  should 
be,  since  the  curve  of  intersection  is  a  parabola,  which  has  no 
asymptote. 

191.  To  develop  this  cone.  Since  every  point  in  the  base  is 
equidistant  from  the  vertex,  the  base  itself  will  develop  into  an 
arc  of  a  circle  whose  radius  is  the  slant  height ;  and  the  length  of 
tliis  arc  will  be  equal  to  the  circumference  of  the  base.  Thus  in 
the  diagram  D,  the  arc  gjc^,  described  mth  radius  j?2^2  —  i^V  5  ^^ 
equal  in  length  to  the  semi-circumference  gfk.  This  semi-circum- 
ference is  bisected  at  /";  and  bisecting  gj{^^  at/*,,  we  have/!,^,  as 
the  position  of  the  element  FP  on  the  developed  sheet :  and  in 
like  manner  the  position  of  any  other  element  may  be  found. 

To  find  the  developed  form  of  the  intersection  :  Lay  oft"    on-  the 


124:  DESCRIPTIVE    GEOMETRY. 

elements  in  the  development,  the  true  distances  from  the  vertex  to 
the  pomts  in  which  these  element  pierce  the  plane  tTt'.  Tims, 
p^u^  =  p'u\  and  p^s^  =  p'z' :  the  element  J^J^'  pierces  the  plane  at 
JV,  but  p'n^  is  a  foreshortened  view  of  a  line  whose  true  length  is 
p'i,  found  by  passing  a  horizontal  plane  through  iT;  therefore  on 
p^/^  make  j)^7i.^  =  p'i^  and  n^  will  lie  on  the  i-equired  curve ;  also 
p^e^  =  p'e" ^  and  so  on. 

To  draw  a  tangent  to  the  developed  curve,  at  any  point  as  n^. 
In  tlie  horizontal  projection  the  tangent  at  n  is  no^  the  hjpothe- 
nuse  of  a  right-angled  triangle yko,  wiiich  triangle  lies  in  the  tan- 
gent plane,  and  will  in  the  development  be  seen  in  its  true  size  and 
form.  Therefore,  drawj^/^,  perpendicular  to  f^p^  and  equal  to  fo\ 
then  o^n^  is  the  required  tangent.  The  principle  is,  that  the  tan- 
gent makes  the  same  angle  with  the  element  after  development  as 
before ;  hence  the  tangents  at  u„  and  z^  are  perpendicular  respec- 
tively to  g^p^  and  \p^. 

192.  To  tind  the  shortest  path  on  the  surface,  between  any  two 
points  as  X  and  Y.  These  points  fall  at  x^  and  y^  in  the  develop- 
ment ;  the  right  line  x^y^  is  the  developed  path ;  it  cuts  pjb^  at  w^ ; 
set  o^p'w"  —  Jp^^i  draw  through  w"  a  horizontal  line  cutting  ^'Z»' 
in  w' ^  the  vertical  projection  of  a  point  in  the  required  curve  :  the 
hoi  izontal  projection  is  best  found  by  setting  oif  pw  equal  to  aw" ; 
and  in  like  manner,  other  points  may  be  determined.  The  highest 
point  M  is  found  by  drawing  p^d.^  perpendicular  to  x^y^ ,  which  it 
cuts  at  m^ :  make  the  2iYQ,fd  equal  to  the  'axcf^d^ ,  and  project  d  to 
d' ;  then  the  position  of  M  on  PD  is  ascertained  as  above. 

Note.  The  problem  of  the  shortest  path,  on  any  surfaces  which 
can  be  developed  either  by  unrolling,  as  in  the  case  of  single-curved 
surfaces,  or  by  unfolding,  like  prisms  or  pyramids,  is  always  solved 
in  this  manner,  provided  that  a  right  line  can  be  drawn  on  the  de- 
veloped sheet  from  one  point  to  the  other,  which,  however,  is  not 
always  possible. 

193.  It  is  clear  that,  as  in  the  case  of  the  cylinder,  the  ellipse 
may  be  turned  end  for  end,  and  the  upper  part  of  the  cone  joined 
to  the  lower  as  shown  in  the  small  diagram  E ;  in  going  from  the 
old  position  to  the  new,  the  vertex  describes  a  semicircle  about  an 
axis  perpendicular  to  the  elliptical  section  at  its  centre. 


DESCKirTlVE    GEOMETRY. 


1^5 


194.  Problem  5.    To  develop  the  helical  convolute. 

Analysis.  If  through  the  tangent  to  the  directrix  at  any  point 
a  plane  be  passed  containing  the  centre  of  curvature  at  that  point, 
tlie  helix  can  be  rolled  upon  that  plane  portion  by  portion ;  and, 
the  radius  of  curvature  being  constant,  it  will  thus  develop  into  a 
circle  having  that  radius.  The  rectilinear  elements  of  the  surface 
being  tangents  to  the  helix,  will  appear  in  the  development  as  tan- 
gents to  this  circle. 

Constrnction.  Let  it  be  required  to  develop  so  much  of  the 
lower  nappe  of  the  convolute  shown  in  Fig.  168,  as  lies  above  H, 


—  ?, 


EiG.  168 


between  the  point  D  and  the  element  GN. 
P 


ture  of  the  helix  is 


cos    CD 


The  radius  of  curva- 
,  in  which  p  is  the  radius  of  the  cylinder 


on  which  the  curve  lies,  and  go  is  the  ohliquity^  or  angle  made  by 
the  tangent  with  a  plane  perpendicular  to  the  axis.  This  may  be 
readily  found  graphically  thus.  The  vertical  projection  of  the  tan- 
gent at  O  is  o'm\  which  cuts  d'f'^  the  outline  of  the  cylinder,  at  c ; 
draw  at  o'  a  horizontal  line,  and  at  c  a  perpendicular  to  o'm  :  these 
intersect  at  A.',  and  o'k  is  the  radius  of  curvature  required. 


126  DESCRIPTIVE    GEOMETRY. 

Then  in  the  diagram  D,  draw  about  any  centre  a?,  a  circle  with 
tliis  radius,  and  on  it  set  off  the  arc  d^o^  g^  equal  to  g'n' ;  this  will 
be  the  development  of  the  helical  arc  DOG.  Since  the  horizontal 
trace  dm  is  the  involute  of  the  original  helix,  its  develoj^ment  will 
be  the  involute  d.r^n^  of  the  arc  dfi^g^^  to  which  g^n^ ,  equal  to  g'n\ 
will  be  tangent. 

195.  The  Problem  of  the  Shortest  Path.  Let  it  be  required  to 
find  the  shortest  path-  on  the  surface,  between  the  points  2>  in  H, 
and  F  on  GJS^.  Tliese  points  fall  in  the  development  at  d^  and  ^, , 
but  they  cannot  be  joined  by  a  right  line  on  the  surface,  since  the 
latter  has  no  existence  within  the  circle.  The  least  distance  is 
.found  by  drawing  from^j  a  tangent  to  the  circle,  and  finding  the 
point  of  tangency  ?/, ;  the  required  line  is,  then,  made  nj)  of  the 
circular  dire  d^u^ ,  and  the  right  line  u^p^. 

The  projections  of  this  path  on  the  original  surface  may  be  de- 
termined as  follows.  In  the  horizontal  projection  bisect  the  quad- 
rant ge  at  a,  and  draw  al  tangent  to  the  circle ;  it  rej)resents  an  ele- 
ment of  which  ajy^  is  the  developed  position,  where  a^  bisects  the 
arc  g^e^ ;  this  cuts  j^j-w,  at  y,.  The  vertical  projection  of  this  point 
must  lie  on  a'l' ;  in  order  to  fix  its  altitude,  set  off  n'y''  =  l^y^ ,  and 
draw  through  y"  a  horizontal  line  cutting  a'V  in  y' .  Project  y"  on 
AB  at  2/2 ;  then  n'y^  will  be  equal  to  Zy,  the  horizontal  projection  of 
l^y^  in  its  original  position.  In  like  manner,  the  location  of  TF  on 
ER  may  be  found,  and  by  drawing  intermediate  elements  any  de- 
sired number  of  points  may  be  determined.  The  restored  position 
of  'Wi  is  best  found  by  setting  off  ou^  the  same  fraction  of  the  quad- 
rant oe  that  o^u^  is  of  the  arc  o.e^.,  and  the  helical  arc  DOUiorm?^ 
the  first  portion  of  the  required  shortest  path. 

The  tangent  to  the  helix  at  U  is  the  element  UZ^  which  in  the 
development  is  i^j^j ,  a  prolongation  oi  p,v,\  and  it  will  be  noted 
that  the  shortest  paths  from  P  to  any  points  on  the  portion  DMZ 
of  the  horizontal  trace  of  the  convolute,  are  equal. 

196.  Problem  6.  To  find  the  intersectionof  a  plane  with  any 
surface  of  revolution. 

Construction.  Every  transverse  section  is  a  circle,  which  in 
general  is  the  simplest  line  that  can  be  drawn  upon  such  a  surface ; 
and  these  circles  are  made  use  of  precisely  as  in  the  case  of  the 


DESCRIPTIVE   GEOMETRY. 


127 


cone.  Tims  in  Fig.  169,  71'  in  the  front  view  represents  a  chord 
in  the  circle  whose  radius  is  cd^  which  chord  is  seen  in  its  true 
length  as  nn  in  the  top  ^-iew  and  as  n^n^  in  the  supplementary  view 


S ;  in'  represents  a  chord  in  the  circle  whose  radius  is  eg^  seen  in 
its  true  length  as  rmn  and  m^m^ ,  and  so  on. 

To  draw  a  tangent  to  the  curve  of  intersection,  at  any  point  as  L. 
Draw  the  tangent  to  the  horizontal  projection  at  l^  by  the  method 
of  Fig.  139.  This  tangent  pierces  the  plane  through  the  axis  par- 
allel to  T,  at  the  point  P ;  of  which  the  supplementary  j)rojection 
is  j9j :  and  Z,^i  is  the  supplementary  projection  of  the  tangent. 

Otherwise :  Draw  a  plane  tangent  to  the  surface  at  L  by  the 
method  of  Fig.  153;  its  intersection  with  the  given  plane  is  the 


128  DESCRIPTIVE   GEOMETRY. 

required  tangent  line,  whose  supplementary  projection  may  be  found 
as  in  Fio^s.  163  and  167. 

197.  In  Fig.  ITO  is  giiown  tlie  "stub  end"  of  a  connecting- 
rod.  It  is  rectangular  in  section,  and  is  joined  to  the  cylindrical 
neck  by  a  surface  of  revolution  whose  contour  is  the  circular  arc 
w'z' ^  described  about  the  centre  K\  w^e  have,  then,  to  find  the  inter- 
sections of  this  surface  by  the  two  planes  tt^  ss,  parallel  to  the  axis. 

A  transverse  section  at  c'  is  a  circle  whose  radius  is  cp' ;  this 
circle  is  seen  in  the  end  view  to  be  cut  by  the  plane  tt  at  d,  which 
is  projected  back  to  d\  a  point  in  the  required  curve ;  and  in  like 
manner  other  points  may  be  determined. 

The  plane  Sii  is  seen  in  the  side  view  to  cut  the  outline  at  Xy 
which  determines  the  vertex  x^'  of  the  curv^e  seen  in  the  top  view. 
A  circle  through  e  is  seen  in  the  end  view  to  cut  the  vertical  centre 
line  in  g,  which,  projected  back  to  the  contour  at  g\  fixes  the  loca- 
tion of  a  transverse  section  from  wdiich  by  the  preceding  ])rocess 
Avould  be  found  the  point  e  in  the  side  view,  corresponding  to  <?"  in 
the  top  view.  A  transverse  section  at  any  point  as  o',  between  x' 
and  g\  is  a  circle  in  which  7in,  in  the  end  view^,  is  a  chord ;  this 
chord  is  also  seen  in  its  true  length  as  7i''n'^  in  the  top  view.  The 
same  circle  is  seen  in  the  end  view  to  be  cut  by  the  plane  U  at  /% 
which  projected  back  to  r'  determines  another  point  in  the  curve 
of  intersection  in  the  side  view.  Other  points  may  be  found  in  a 
similar  manner. 

198.  Tangents  to  the  Curves  of  Intersection.  By  applying  the 
method  of  (175),  a  tangent  may  be  drawn  to  either  of  these  curves 
at  any  point,  with  one  exception. 

It  is  seen  in  the  end  view  that  the  circle  through  w  is  tangent 
to  the  plane  tt,  at  the  point  y ;  and  the  intersection  of  this  plane 
with  the  surface,  as  seen  in  the  side  view,  consists  of  two  sym- 
metrical branches,  which  intersect  at  y\  The  tangent  plane  at  this 
point  coincides  with  the  cutting  plane ;  consequently  the  tangent 
line  cannot  be  determined  by  the  usual  process.  But  a  tangent  at 
this  point  to  the  lower  branch  may  be  constructed  as  follows : 

The  given  centre  of  curvature  at  w'  is  the  point  A";  produce 
Jvw'  to  /,  bisect  KI  at  O,  and  about  0  describe  a  semicircle  on  7i'/ 
as  a  diameter,  cutting  tlie  projection  of  the  axis  at  J^Z     On  7i/  set 


DESCKIPTIVE    GEOMETRY. 


129 


-30  ^  DESCRIPTIVE    GEOMETRY. 

o^y' G  =^  y'F\  also  about  y'  as  a  centre  describe  a  circular  arc 
with  radius  y'w' :  from  G  draw  a  tangent  to  this  arc,  then  ylj<) 
perpendicular  to  that  line,  is  the  required  tangent  at  y' . 

This  construction  is  based  upon  the  consideration  that  if  the 
contour,  w'z\  be  the  osculating  circle  of  an  hyperl)ola  of  which  w' 
is  the  vertex  and  y'  the  centre,  then  y'L  thus  determined  will  be 
one  of  tlie  asymptotes.  Had  that  hyperbola  been  the  actual  contour, 
the  section  of  the  surface  by  the  plane  tt^  as  will  subsequently  be 
shown,  would  have  been,  not  a  curve,  but  the  right  line  y'L  itself. 

INTERSECTIONS    OF    SINGLE- CURVED    SURFACES. 

199.  In  Fig.  171,  draw  any  line  otI'  through  the  vertex  o  of 
the  cone ;  it  pierces  the  plane  of  the  base,  M^  at  the  point  n.  Draw 
in  the  plane  M  any  line  nc  through  the  ])oint  n^  cutting  the  base  at 
€  and  d\  it  is  clear  that  the  plane  one  cuts  from  the  cone  the  two 
elements  oc,  od.  Draw  in  the  plane  J/^  through  n^  a  line  tangent 
to  the  base  at  / ;  then  the  plane  onl  ig  tangent  to  the  cone  along  the 
element  ol.  And  the  like  will  be  true  of  any  other  cone  whose 
base  is  in  the  same  plane  J/,  if  its  vertex  also  lies  in  the  same  line 
on.  It  is  now  proposed  to  apply  this  in  the  solution  of  the  follow- 
iac:  problem. 

200.  ir*KOBLEM  1.  To  find  the  intersection  of  two  cones  whose 
bases  are  in  the  same  jpland. 

Analysis.  Pass  a  series  of  planes  through  both  vertices.  These 
will  cut  elements  from  each  cone ;  and  the  intersections  of  those 
cut  from  one  cone  with  those  cut  from  the  other  will  be  points  in 
the  required  curve. 

Construction.  Since  the  bases  are  in  the  same  plane,  the  two 
cones  may  be  so  placed  that  this  plane  shall  coincide  with  H,  as  in 
Fig.  172.  Join  the  vertices,  P  and  0^  by  a  right  line,  and  pro- 
duce it  to  pierce  H  in  ]^.  Then  as  in  Fig.  171,  nc  is  the  horizon- 
tal trace  of  a  plane  which  contains  the  line  T^iT,  and  this  plane  cuts 
from  the  cone  X  two  elements  whose  vertical  projections  are  c'o' 
and  d'o' .  The  trace  nc  also  cuts  the  base  of  the  cone  l"in  e,  ver- 
tically projected  at  e  in  AB ;  and  e'p\  the  vertical  projection  of  the 
element  cut  from  that  cone,  intersects  cVand  d'o'  in  the  points  h' ^ 
g'^  which  therefore  lie  on  the  vertical  projection  of  the   required 


DESCRIPTIVE   GEOMETRY.  131 

curve :  in  a  similar  maimer  any  desired  number  of  points  may  be 
determined. 

201.  In  the  construction  of  such  curves,  there  are  certain  criti 
cal  and  limiting  points  which  it  is  always  desirable  to  locate.  For 
instance,  o'f  is  the  extreme  visible  element  on  the  left,  in  the  ver- 
tical projection  of  the  cone  X.  And,  just  as  in  Fig.  Vl\^  fn  is  the 
horizontal  trace  of  a  plane  containing  that  element  and  the  line 
0N\  tliis  plane  cuts  from  the  cone  Y  an  element  whose  vertical 
projection  i^'jp'  intersects  o'f  in  a  point  s' :  and  at  this  point  o'f  is 
tangent  to  the  vertical  projection  of  the  curve.  The  points  of  con- 
tact on  the  right-hand  element  of  X,  and  on  the  left-hand  element 
of  y,  are  found  in  a  similar  manner. 

Draw  nl  tangent  to  the  base  of  X\  it  is  the  horizontal  trace  of 
a  plane  tangent  to  that  cone  along  the  element  whose  vertical  pro- 
jection is  I'o' ;  this  plane  cuts  from  Y  an  element  whose  vertical 
projection  mp'  is  tangent  to  the  vertical  projection  of  the  curve  at 
h' ^  its  intersection  with  I'o' .  And  another  tangent  may  be  deter- 
mined by  means  of  another  tangent  plane,  whose  horizontal  trace  is 
nx. 

202.  Attention  has  thus  far  been  purposely  confined  to  the 
vertical  projection  of  the  curve.  The  horizontal  projection,  of 
course,  will  be  determined  by  the  intersections  of  the  horizontal 
projections  of  the  elements ;  but  the  effect  of  introducing  these,  in 
a  diagram  upon  so  small  a  scale,  and  necessarily  involving  so 
many  other  lines,  would  have  been  simply  bewildering,  for  which 
reason  they  have  been  omitted.  In  constructing  that  projection, 
it  will  be  found  advantageous  to  pass  auxiliary  planes  through  the 
extreme  visible  elements  of  the  cones  in  the  top  view  also,  in  order 
to  locate  the  points  at  which  the  curve  appears  tangent  to  those 
elements. 

It  is  also  to  be  noted,  that  the  two  projections  of  these  curves 
arc  in  a  sense  independent  of  each  other.  Thus,  the  bases,  and 
the  vertical  projections  of  the  cones,  remaining  as  they  are,  the 
horizontal  projections^,  o,  of  the  vertices  may  lie  upon  any  oblique 
line  drawn  through  n.  AYhence  it  appears  that  an  infinite  number 
of  pairs  of  cones  may  be  constructed  having  the  same  bases  and 
altitudes,  whose  curves  of  intersection  will  all  have  the  same  verti- 


132  DESCRIPTIVE    GEOMETRY. 

cal  projection.  In  a  similar  manner  it  may  "be  sliown  that  differ- 
ent pairs  of  cones  may  intersect  in  curves  wliicli  have  the  same 
horizontal  projection. 

203.  Fig,  172  represents  a  case  of  complete  interpenetration ; 
the  cone  A"  enters  the  cone  Y  on  the  left,  passes  bodily  through  it, 
and  emerges  on  the  right,  thus  forming  two  distinct  curves  of  in- 
tersection. This  is  indicated  in  the  horizontal  projection  by  the 
jircumstance  that  every  one  of  the  horizontal  traces,  inchiding  the 
two  which  are  tangent  to  the  base  of  A"^,  cuts  the  base  of  Y  in  two 
points.     Each  of  the  auxiliary  planes,  then,  cuts  two  elements  from 

Y,  although  it  may  contain  only  one  element  of  X;  l)ut  the  latter 
will  intersect  both  the  former;  thus,  I/o'  cuts  7)i'j/  at  k\  and  it 
also  cuts  ap'  at  q\  wdiich  is  the  vertical  projection  of  the  point  of 
tangency  between  the  second  element  and  the  other  curve. 

204.  Now  suppose  the  cone  l^to  be  revolved  around  a  verti- 
cal line  through  P,  in  the  direction  indicated  by  the  arrow.  The 
jDoints  a  and  m,  and  consequently  the  points  q'  and  /•',  will  ap- 
]3roach  each  other,  until  when  the  base  becomes  tangent  to  l/i  as 
shown  by  the  dotted  line,  the  auxiliary  ])lane  will  be  taiigerit  to 
both  cones.  The  two  curves  will  then  coalesce,  forming  one  con- 
tinuous line,  which,  as  shown  in  Fig.  173,  will  crusts  itself  at  the 
intersection  oi  LO  andZP,  the  two  elem^ents  of  contact,  not  being 
tangent  to  either  of  them. 

The  interpenetration  is  still  complete ;   but  the  exact  limit  has 
now  been  reached,  and  if   the   revolution   of   Y  be   carried  any 
farther,  this  will  no  longer  be  the  case.      The  plane  tangent  to  X 
along  the  element  Z  0  will  then  pass  outside  of  the  cone  Y\   no 
part  of  either  surface  will  be  entirely  buried  within  the  other,  and 
the  two  will  intersect,  as  shown  in  Fig.  174,  in  a  continuous  curve 
which  does  not  cross  itself.     In  this  diagram  the  cones  are  of  equal 
altitude ;   the  line  joining  the  vertices  is  therefore  parallel  to  H,  and 
for  convenience  it  is  here  made  parallel  to  V  also,  so  that  the  traces 
of  the  auxiliary  planes  are  parallel  to  AB.      The  auxiliary  plane 
tangent  to  Y  along  the  nearest  element  77^,  cuts  from  X  the  two 
elements  JO^  LO,  which  are  tangent  to  the  curve  at  the  jjoints  C, 
D,  in  which  they  intersect  IP.      Sijnilarly,  an  auxiliary  plane  tan- 
gent to  A^  along  its  most  remote  element,  will  cut  from  l^two  ele- 


DESCRIPTIVE   GEOMETRY. 


133 


mcnts,  both  of  wliicli  are  tangent  to  the  curve ;  and  other  points 
are  found  as  previously  explained. 


205.  If  the  cones  as  ^iven  hare  not  a  common  base,  a  very  obvi- 
ous expedient  is  to  provide  one,  by  passing  a  plane  which  cuts  all 
the  rectilinear  elements  of  both.  This,  however,  is  not  absolutely 
necessary  nor  always  even  desirable.  The  base  of  Y^  Fig.  175,  be- 
ing in  the  horizontal  plane,  it  is  clearly  always  possible  so  to  place 
the  two  cones  that  the  base  of  X  shall  lie,  as  liere  shown,  in  a  plane 
tTt\  perpendicular  to  V.  Drawing  OP,  produce  it  to  pierce  H  in 
iV^,  and  the  other  plane  in  M.  Revolve  tTt'  into  Y ;  then  the  base 
of  X  will  be  seen  in  its  true  form  and  in  its  correct  position  rela- 
tively to  the  point  J/",  w^hich  falls  at  in" .  All  the  traces  of  the 
auxiliary  planes  upon  tTt\  then,  will  pass  through  m\  and  all 
their  horizontal  traces  through  n.  Draw  "rri"/",  tangent  to  the 
base  at  e' ;  this  is  the  trace  of  a  plane  tangent  to  X  along  the  ele  • 
ment  EO.  Set  off  Tf  =  Tf\  and  draw  nf\  this  is  the  horizontal 
trace  of  the  same  plane,  which  is  now  perceived  to  cut  from  l^tlie 
two  elements  CP^  DP :  these  cut  EO  in  two  points  of  the  curves 
sought. 


134 


DESCRIPTIVE   GEOMETRY. 


The  construction,  in  short,  is  effected  precisely  as  in  the  pre- 
ceding cases :  for  instance,  in  order  to  find  a  point  on  IIP  the  ex- 
treme right-hand  element  of  Y^  draw  nrg  the  horizontal  trace  of 
an  anxiharj  plane  containing  that  element,  set  off  Tg"  —  Tg^  and 


draw  g^'m"  cutting  the  base  of  X  at  s" ;  project  s''  to  s\  then  s'o' 
is  the  vertical  projection  of  an  element,  cutting  r'^'  in  z\  which  is 
the  vertical  projection  of  the  required  point. 

206.  Manipulation  in  Construction — It  is  a  very  common  mis- 
take to  suppose  tliat  in  making  such  constructions  as  those  of  Figs. 
172  and  175,  time  may  be  saved  by  drawing  at  once  both  projec- 
tions of  a  great  number  of  elements,  and  then  selecting  those 
which  intersect  in  the  required  points.  ■  But  the  effect  of  such  a 
maze  of  lines  is  bewildering,  and  the  attempt  is  more  than  likely 


DESCRIPTIVE    GEOMETRY. 


135 


to  result  in  error,  confusion,  and  absolute  loss  of  time.  A  much 
safer  and  on  many  accounts  better  course  is  to  find  first  a  point 
situated  for  instance  on  tlie  nearest  element  of  one  of  the  surfaces, 
and  then  to  follow  the  curve  around  point  by  point  in  one  direc- 
tion, sketching  it  in  lightly  as  each  additional  point  is  located.  In 
determining  a  point,  do  not  draw  in  the  trace  of  the  auxiliary 
plane,  but  simply  mark  where  it  cuts  the  bases ;  and  do  not  draw 
the  w^iole  projection  of  either  element,  because  ordinarily  a  short 
bit  of  one  will  sufiice,  upon  which  is  marked  the  point  in  which  the 
other  cuts  it. 

207.  If  the  cones  are  so  situated  as  to  have  two  auxiliary  tan- 
gent planes  in  common,  as  in  Fig.  176,  then  it  is  clear  that  all  the 


Fig.  176 


elements  of  both  surfaces  will  be  cut,  and  that  the  line  of  intersec- 
tion will  cross  itself  at  two  points  on  opposite  sides  of  the  cones. 
The  line  may  be  of  double  curvature;  but  if  the  bases  of  both  cones 
are  conic  sections,  it  will  in  general  be  composed  of  two  plane  curves, 

which  tlierefore  are  themselves  conic  sections. 

An  exceptional  case  is  illustrated  in  Fig.  177,  where  J9,  o,  are 


136 


DESCRIPTIVE    GEOMETRY. 


the  vertices  of  two  similar  cones  of  revolution  of  equal  altitude, 
whose  axes  are  vertical  and  therefore  parallel. 


The  extreme  elements  jpr^  os^  intersect  in  A ;  an  auxiliary  plane 
through  po  cuts  from  one  cone  tlie  elements  pa^  pb^  and  fi'om  the 
other,  the  elements  oe^  of:  pa  cuts  oe  in  n^  but  it  is  parallel  to 
of^  and  oe  is  parallel  to  ph.  Consequently  these  lower  nappes  of 
the  cones  can  intersect  in  only  one  line,  the  hyperbola  kng\  the 
upper  nappes  obviously  intersect  in  the  opposite  branch  of  the 
same  curve :  and  it  is  quite  evident  that  a  similar  state  of  tilings 
may  -exist  with  cones  of  other  forms. 

In  Fig.  178,  the  angles  at  the  vertices  are  the  same  as  in  Fig. 
177,  and  the  vertices  are  equidistant  from  g^  the  intersection  of  the 


Fig.  178 


Fig.  179 


axes,  whose  inclination  is  such  that  the  extreme  elements  j^^/,  oii-.^ 
ai*e  not  parallel.     In  this  case  pa  not  only  cuts  oe  in  71^  but  it  cuts 


DESCRIPTIVE    GEOMETRY.  137 

of  in  h^  and  oe  cuts  ph  in  m ;  thus  tlie  lower  nappes  intersect  in 
the  Jiyperbola  ling^  and  also  in  the  ellipse  seen  edgewise  as  the 
right  line  cd :  the  upper  nappes  intersect  in  the  opposite  branch  of 
the  same  hyperbola. 

In  Fig.  179  the  same  cones  are  show^n,  the  vertices  equidistant 
from  i  the  intersection  of  the  axes,  but  so  inclined  that  jpy  and  ow 
are  parallel.  In  these  circumstances  the  surfaces  intersect  in  the 
parabola  Idg^  and  also  in  the  ellipse  cd ;  the  upper  nappes  do  not 
meet  at  all.  The  same  cones,  it  is  clear,  can  be  so  placed  as  to  in- 
tersect each  other  in  two  ellipses,  as  do  those  shown  in  Fig.  176 : 
in  which  case  also  the  opposite  nappes  do  not  intersect. 

208.  In  the  case  of  any  two  given  cones,  the  question  whether 
there  will  be  any  line  of  intersection  wdth  infinite  branches  may 
be  decided  thus:  Draw  through  the  vertex  of  either  a  series  of 
lines  parallel  to  the  elements  of  the  other;  then  in  general  this 
third  cone  will  either  have  only  the  vertex  in  common  with  the 
first,  or  be  tangent  to  it  along  one  element,  or  cut  it  in  two  ele- 
ments. 

In  the  first  case,  no  element  of  either  of  the  given  cones  will 
be  parallel  to  any  element  of  the  other,  and  the  intersection  will 
consist  of  one  or  two  closed  curves;  which,  w^hether  of  double 
curvature  or  not,  belong  by  analogy  to  the  class  of  ellvptic  inter- 
sections. 

In  the  second  case,  one  element  of  each  cone  will  be  parallel  to 
one  element  of  the  other ;  and  there  will  be  one  line  of  intersection 
consisting  of  an  infinite  branch  with  no  asymptote:  which  there- 
fore belongs  to  the  class  of  parabolic  intersections. 

In  the  third  case,  two  elements  of  each  cone  will  be  parallel  to 
elements  of  the  other ;  the  surfaces  will  intersect  in  two  infinite 
branches  with  asymptotes,  which  belong  by  analogy  to  the  class  of 
hyperholic  intersections:  and  the  asymptotes  are  determined  by 
drawing  ])lanes  tangent  to  the  cones  along  the  parallel  elements; 
these  will  intersect  in  the  required  lines. 

In  the  last  two  cases  there  will  be  one  closed  curve  of  intersec- 
tion, besides  the  infinite  ones.  But  should  the  two  cones  have  all 
their  elements  respectively  parallel,  as  in  Fig.  177,  there  will  be  an 
intersection  composed  of  two  infinite  branches,  and  no  other.     The 


138  DESCllIPTlYE    GEOMETRY. 

cones  in  tliis  case  have  necessarily  two  common  tangent  planes,  and 
the  elements  of  contact  are  the  ones  to  which  the  asymptotes  will 
be  parallel.  If  the  intersection  is  of  single  curvature,  the  asymp- 
totes will  be  determined  by  the  intersections  of  the  plane  of  the 
curve  with  the  two  tangent  planes ;  but  if  not,  the  asymptote  can- 
not be  graphically  determined :  because  the  planes  tangent  to  the 
two  cones  along  the  parallel  elements  now  coincide,  giving  no  line 
of  intersection. 

209.  If  one  of  the  cones  becomes  a  cylinder,  the  intersection  may 
be  found  in  substantially  the  same  manner.  Draw  a  line,  parallel 
to  the  elements  of  the  cylinder,  through  the  vertex  of  the  cone ; 
then  auxiliary  planes  through  this  line  will  cut  from  both  surfaces 
rectilinear  elements,  whose  intersections  will  be  i^oints  in  the  re- 
quired curve.  Draw  such  planes  tangent  to  each  surface ;  then, 
as  before,  if  both  planes  tangent  to  either  cut  the  other,  there  will 
be  two  separate  curves ;  if  one  plane  tangent  to  each  cut  the  other, 
there  will  be  one  continuous  curve ;  if  tliere  be  one  common  tan- 
gent plane  the  curve  wdll  cross  itself  once ;  and  if  two,  it  will  cross 
itself  twice,  and  will  consist  of  two  conic  sections  when  the  base  of 
each  surface  is  a  conic  section. 

Infinite  Intersections Since  all  the  elements  of  the  cylinder 

are  parallel  to  the  auxiliary  line  through  the  vertex,  they  cannot 
be  parallel  to  any  element  of  the  cone,  unless  that  line  itself  lies  in 
the  conical  surface ;  if  it  does,  there  may  be  curves  of  intersection 
with  infinite  branches.  Draw  a  test  plane  tangent  to  the  cone 
along  this  auxiliary  line ;  it  will  either  pass  outside  the  cylinder,  or 
be  tangent  to  it,  or  cut  it  in  two  elements. 

In  the  first  case,  the  intersection  will  be  a  single  closed  curve. 

In  the  second  case,  it  will  in  general  consist  of  one  infinite 
branch  with  a  single  asymptote. 

In  the  third  case  it  will  in  general  consist  of  two  infinite 
branches  with  two  common  asymptotes. 

210.  That  these  things  are   so,  is  very  clearly  shown  by  so 
placing  the  surfaces  that  the  auxiliary  line  is  vertical.     Thus  in  Fig. 
180,  which  is  a  horizontal  projection,  let  o  represent  this  line;   let 
M  be  the  horizontal  trace  of  the  cone,  N  that  of  the  cylinder,  TT 
that  of  the  test  plane,  and  ox^  oy^  oz,  those  of  auxiliary  planes. 


DESCRIPTIVE    GEOMETRY. 


139 


Each  auxiliary  plane  contains  the  auxiliary  line,  which  is  parallel  to 
the  elements  of  the  cylinder ;  but  it  also  cuts  from  the  cone  another 
element  which  is  not  parallel  to  them,  and  since  M  is  in  the  hori- 
zontal plane,  while  the  vertex  is  above  it,  it  is  apparent  that  in  this 


Fig.  180 


case  every  element  of  the  cylinder  must  pierce  the  cone  in  a  point 
of  the  lower  nappe.     Had  the  cylinder  been  placed  on  the  opposite 
side  of  TT^  the  intersection  would  have  lain  on  the  upper  nappe. 
Fig.  181  illustrates  the  second  case,  the  test  plane  being  tangent 


Fig.  181 


to  N\  here  it  is  evident  that  as  the  auxiliary  plane  approaches 
coincidence  with  TT^  the  elements  cut  from  the  cylinder  approach 
the  line  of  contact  a^  while  the  one  .cut  from  the  cone  becoming 
more  nearly  parallel  to  them,  meets  them  at  points  more  and  more 
remote,  receding  to  infinity  at  the  limit.  Consequently  the  curve 
of  intersection  lies  wholly  on  the  loM-er  nappe,  and  the  element  a 
of  the  cylinder  is  an  asymptote  to  both  sides  of  the  single  iniinite 
branch,  and  lies  between  them,  but  nearer  to  one  than  to  the  other. 
An  exception  to  this  occurs  when,  as  in  Fig.  182,  the  two  sur- 
faces are  tangent  along  an  element.      Let  oy  be  the  trace  of  any 


140 


DESCllIPTLVE    GEOMETRY. 


auxiliary  plane,  cutting  Jf  in  r  and  JV  in  d:  then  ot^  is  tlie  projec- 
tion ox  that  part  of  an  element  of  the  cone  included  between  the 
vertex  and  the  base,  and  od  the  projection  of  that  part  included 
between  the  vertex  and  the  line  of  intersection.  The  length  of  the 
former  being  always  finite,  that  of  the  latter  must  also  be  finite ; 
therefore  the  surfaces  intersect  in  a  single  closed  curve. 

211.  The  third  case  is  illustrated  in  Fig.  183;  by  comparison 
with  Fig.  181,  it  is  obvious  that  the  part  of  the  cylinder  mcluded 
between  TT'and  the  tangent  auxihary  plane  02,  will  intersect  the 
cone  in  an  infinite  branch  on  the  lower  nappe,  to  which  the  two 


Fig.  183 


Fig.  184; 


elements  a  and  h  of  the  cylinder  will  be  asymptotes.  Also  that  the 
portion  on  the  opposite  side  of  the  test  plane  will  give  another 
infinite  branch  on  the  upper  nappe,  liaving  as  asymptotes  the  same 
elements  a  imd  b. 

There  will  be  an  exception  to  this  also,  when  the  two  surfaces 
intersect  in  the  auxiliary  line,  as  in  Fig.  184.  They  will  also  inter- 
sect in  one  continuous  curve,  w^iich,  crossing  the  common  element 
at  the  vertex,  extends  both  ways  to  infinity,  and  is  asymptotic  in 
each  direction  to  the  other  element  h  cut  from  the  cylinder  by  the 
test  plane, 

212.  In  the  preceding  arguments  relating  to  tlie  nature  and  the 
number  of  the  curves  of  intersection,  it  has  been  assumed  that,  as 
is  usual  in  practical  cases,  the  cones  and  cylinders  have  closed 
mirves  as  bases,  and  are  externally  convex  throughout.  Almost 
endless  variations  miglit  result  from  constructing  one  or  both  of  the 


DESCRIPTIVE    GEOMETRY.  141 

surfaces  with  spiral,  sinuous,  or  infinite  curves  for  bases;  these 
liowever  it  is  not  proposed  to  consider,  since  they  involve  no  prm- 
ciple  which  has  not  already  been  explained. 

213.  Intersection  of  Two  Cylinders.  The  same  general  method 
is  still  applicable :  Draw  a  plane  parallel  to  the  elements  of  each 
cylinder ;  then  auxiliary  planes  parallel  to  this  plane  will  cut  from 
both  surfaces  rectilinear  elements,  and  their  intersections  will  be 
points  in  the  required  curve.  By  drawing  two  such  planes  tangent 
to  each  surface,  the  questions  as  to  whether  there  will  be  one  or  two 
curves,  one  or  two  crossings,  etc. ,  are  decided  exactly  as  in  (208) ; 
and  w4ien  there  are  two  common  tangent  planes,  the  intersection 
will  consist  of  two  conic  sections,  if  the  bases  themselves  are  conic 
sections. 

As  to  infinite  intersections,  it  is  perfectly  obvious  that,  if  the 
bases  are  closed  curves,  there  can  be  none,  unless  the  elements  of 
one  cylinder  are  parallel  to  those  of  the  other ;  w^hen  they  will 
intersect  in  either  two  or  four  elements,  if  at  all.  But  any  infinite 
curve  can  be  made  the  common  base,  and  therefore  the  line  of 
intersection,  of  two  cylinders. 

214.  Other  Methods  of  Operationc  It  appears  then  from  the 
foregoing,  that  if  two  cones,  two  cylinders,  or  a  cone  and  a  cylinder, 
intersect  each  other,  it  is  edwajs  possible  to  employ  a  system  of  aux- 
iliary planes  which  cut  rectilinear  elements  from  each  surface.  But 
it  must  not  be  inferred  that  this  is  the  only  or  always  even  the  most 
eligible  expedient.  In  probably  the  greater  number  of  practical 
cases,  these  surfaces  have  circular  bases ;  and  when  this  is  so,  planes 
which  cut  circles  from  each,  or  right  lines  from  one  and  circles 
from  the  other,  can  often  be  employed  to  great  advantage. 

Thus  in  Fig.  185,  the  horizontal  plane  i'f  cuts  from  the  cone 
a  circle,  which  in  the  horizontal  projection  is  seen  to  pierce  the 
cylinder  at  e  and  n,  which  are  vertically  projected  to  e,  n' ;  and 
any  number  of  points  may  be  found  in  like  manner. 

Draw  ox^  oy^  tangent  to  the  base  of  the  cylinder;  describe 
through  r  and  <?,  the  points  of  contact,  a  circle  about  o,  cutting  ow 
in  6?,  whose  vertical  projection  is  d' '^  then/'',  c\  must  lie  on  a 
horizontal  plane  thrpugh  d'^  and  at  those  points  the  curve  is  tangent 
to  o'x\  o'y' .     Draw  through  a^  the  centre  of  the  base  of  the  cyl- 


142 


DESCRIPTIVE   GEOMETRY. 


inder,  the  right  line  ov ;  it  cuts  tlie  base  in  s  and  Ic,  whose  vertical 
projections,  found  in  the  same  manner  as  r'  and  c\  are  the  highest 
and  lowest  points  of  the  intersection. 


FiGf.  185 


215.  Only  the  front  half  of  the  cone  is  represented  in  Fig.  185  ; 
and  in  Fig.  186  is  given  the  development  of  that  half,  showing  the 
form  of  the  hole  which  must  be  cut  in  tlie  sheet  for  the  insertion 
of  the  cylinder.  The  base  develops  into  the  arc  of  a  circle  whose 
radius  o^z^^  is  equal  to  o'z\  the  slant  height  of  the  cone,  and  ^he 


DESCRIPTIVE    GEOMETRY.  143 

length  of  tlie  arc  z^ii^w^  is  equal  to  that  of  the  seniicircumference 
zuii\  Make  the  arc  it{i\  =  arc  tcv,  set  off  the  arcs  ^',a?, ,  v,y^,  equal 
to  the  arc  vx,  and  draw  o.y,,  o^v ,,  o,x^.  Describe  about  6>,  an  arc 
with  radius  o,  J, ,  equal  to  o'd\  thus  locating  the  points  of  tangency 
r^ ,  c\  ;  describe  about  o^  an  arc  with  radius  o^fi  =  o'/'',  cutting  o^v^ 
in  m, ,  set  off  the  arcs  m^n^ ,  m,e^ ,  each  equal  to  me,  then  e^ ,  n^ , 
lie  on  the  developed  curve :  and  other  points  may  be  found  in  a 
similar  manner.  The  vertices,  s^  and  A;, ,  are  determined  by  setting 
off  on  o^v^  the  true  distances  of  the  points  S  and  ^from  the  vertex 
of  the  cone,  which  are  respectively  equal  to  o^t\  o'V . 

•  216.  The  construction  of  Fig.  187  may  be  explained  thus: 
The  point  in  which  any  line  on  either  the  horizontal  or  the  incliae(? 
cylinder  pierces  the  vertical  one,  is  seen  directly  in  the  top  view, 
Thus,  the  nearest  element  lik  of  the  horizontal  cylinder  cuts  the  cir- 
cumference at  ^,  which  is  projected  vertically  to  h'  in  the  front 
view :  any  point  c"  on  the  circumference  in  the  end  view  repre- 
sents an  element,  seen  as  c'd'  in  the  front  view,  and  as  cd  in  the  top 
view,  where  the  distance  xc  from  the  centre  line  ie  equal  to  x"g"  ; 
and  d'  is  vertically  over  d.  Similarly,  the  nearest  element  of  the 
inchned  cylinder  is  represented  by  m'o'  in  the  front  view  and  by 
"iiio  in  tlie  top  view,  and  the  altitude  of  the  point  in  which  it 
pierces  the  upright  cylinder  is  determined  by  projecting  o  up  to  o' . 
A  supplementary  view  looking  in  the  direction  of  the  arrow  shows 
the  base  of  the  inclined  cylinder  in  its  true  form,  and  any  point  n^ 
upon  its  circumference  represents  an  element  of  which  the  front 
view  is  nr'  and  the  top  view  is  nr,  located  by  making  wn  =  w^n^ ; 
and  t'  is  vertically  over  r :  any  number  of  points  in  the  curves  may 
he  found  in  the  same  manner.  In  this  way  the  determination  of 
the  intersections  might  be  made  clear  to  one  totally  unfamiliar  with 
the  stage  machinery  of  Descriptive  Geometry :  though  it  is  very 
evident  that  the  operations  are  actually  equivalent  to  the  use  of  a 
sci-ies  of  horizontal  planes  in  the  first  case  and  a  series  of  vertical 
ones  in  the  second. 

217.  Just  such  cases  as  these  last  are  the  ones  most  often  met 
with  in  practice ;  and  accordingly,  they  are  the  ones  most  seldom 
illustrated  in  theoretical  treatises  on  the  principles  involved  in  them. 
In  the  appHcation  to  sheet-metal  work,  the  development  is  of  im^ 


144 


DESCRIPTIVE   GEOMETRY. 


portance;  and  that  of  tlie  iipriglit  cylinder  is  given  in  Fig.  188, 
Supposing  it  to  be  cut  vertically  along  tlie  most  remote  element  u, 
and  unrolled  to  right  and  left,  the  surface  will  form  a  sheet  of  a 
breadth  equal  to  the  height  of  the  cylinder,  and  a  length  equal  to 
its  circumference.     The  intersection  with  the  horizontal  cylmder 


Fig.  188 

will  develop  into  a  curve  symmetrical  about  a  horizontal  line  whose 
distance  from  the  lower  edge  is  equal  to  that  of  the  axis  from  AB 
in  Fig.  187.  Rectify  the  arcs  ul^  Ik^  in  the  top  view,  in  the  de- 
velopment set  off  the  distances  ul^  Ik^  equal  to  them,  and  subdivide 
the  latter  into  parts  respectively  equal  to  the  partial  arcs  M,  dfy 


DESCllTPTIVE    GEOMETRY. 


145 


etc.  ;  at  the  points  of  subdivision  erect  vertical  ordinates  equal  to 
the  distances  of  d' ^  f\  etc. ,  above  the  centre  line  of  the  horizontal 
cylinder  in  the  front  view ;  the  required  curve  passes  througli  the 
extremities  of  these  ordinates.  The  intersection  witli  the  inclined 
cylinder  will  develop  into  a  curve  symmetrical  about  22^  the  posi- 
tion of  tlie  right-hand  element  of  the  upright  cylinder  on  the  un- 
rolled slieet;  the  altitudes  of  the  vertices,  <^',  J,  are  taken  directly 
from  the  front  view  :  on  zz  mark  also  the  points  1,  2,  3,  at  altitudes 
equal  to  the  distances  of  s\  0%  /,  above  AB,  and  at  these  points 
draw  horizontal  ordinates  1^,  2(9,  3^%  respectively  equal  in  length 
to  the  rectified  arcs  sz^  oz^  rz^  in  the  top  view :  any  desired  mimber 
of  points  may  be  found  in  a  similar  manner. 


INTERSECTIONS    OF    DOCBLE-CURVED    SURFACES. 

218.  Problem  1.  To  find  the  intersection  of  two  surfaces  of 
revolutioru  whose  axes  are  in  the  same  plane. 

Analysis.  If  the  axes  intersect,  take  the  point  of  intersection 
as  the  common  centre  of  a  series  of  auxiliary  spheres.  Each  spliere 
will  cut  a  circle  from  each  of  the  given  surfaces  (137) ;  the  circum- 
ferences of  these  circles  will  cut  each  ather  in  two  points,  which  lie 
upon  the  required  curve.  If  the  axes  are  parallel,  the  spheres  be- 
come planes  perpendicular  to  the  axes. 

Construction.  In  the  side  view  at  the  left,  Fig.  189,  the  plane 
of  the  axes,  which  is  j^arallel  to  the  paper,  contains  the  visible  coii- 


Fia.  189 


tours  of  the  given  surfaces  and   of  the    spheres  whose  common 


146  DESCRIPTIVE    GEOMETRY. 

centre  is  w ;  tlie  intersections  of  tlie  former,  at  /'  and  s,  give  at  once 
two  points  of  the  required  curve.  A  spliere  tangent  to  one  surface 
around  the  circle  oy^  cuts  the  otlier  in  the  circle  ^a?,  and  oy  cuts^a? 
in  n ;  another  and  larger  sphere  cuts  the  first  surface  in  circles 
through  d  and  g^  and  the  second  in  a  circle  through  c^  and  the  in- 
tersections of  these '  circles  at  e  and  h  also  lie  upon  the  required 
curve  snr :  any  desired  number  of  points  may  be  determined  in  like 
manner. 

In  the  end  view,  the  points  s,  r,  are  projected  directly  to  s^,  /\, 
on  tlie  vertical  centre  line.  The  point  7i  in  the  side  view  representis 
a  chord  in  the  circle  oy,  whos6  circumference  being  seen  in  its  true 
form  and  size  in  the  end  view,  the  points  n^ ,  n^.,  are  found  by  pro- 
jecting n  across  to  that  circumference;  and  ^,,  /^,,  are  located  in  a 
similar  manner. 

Note.  This  method  is  equally  applicable  to  the  case  of  single- 
curved  surfaces  of  revolution  whose  axes  intei'sect.  Its  application 
in  the  extreme  case  where  the  intersection  is  infinitely  remote  and 
the  spheres  become  planes,  has  already  been  illustrated  in  finding 
the  intersection  of  a  cone  with  a  cylinder.  Fig.  185. 

219.  If  the  axes  lie  in  different  planes  the  determination  of  the 
intersection  of  the  surfaces  is  in  general  much  more  laborious.  In 
most  cases,  it  would  probably  be  advisable  to  use  a  system  of  auxiliary 
planes  perpendicular  to  one  of  the  axes,  thus  cutting  circles  from 
the  surface  to  which  that  axis  belongs ;  but  it  still  remains  to  deter- 
mine the  form  of  the  line  cut  from  the  other  surface  by  each  in- 
dividual plane. 

220.  Problem  2.  To  find  the  intersection  of  any  ohlique  cone 
with  a  sphere. 

Analysis.  Pass  a  series  of  auxiliary  planes  through  the  vertex ; 
each  will  cut  a  circle  from  the  spliere  and  two  elements  from  the 
cone ;  and  the  intersections  of  these  elements  with  the  circumference 
of  that  circle  will  be  points  in  the  required  curve. 

Construction.  In  Fig.  190,  6''  is  the  centre  of  the  sphere,  0  i? 
the  vertex  of  the  cone,  whose  base  for  the  sake  of  convenience  is 
placed  in  H ;  and  the  auxiliary  planes  are  vertical.  Let  jpp,  drawn 
at  pleasure  through  o,  be  the  horizontal  trace  of  one  of  these  planes, 
cutting  the  base  of  the  cone  in  d  and  r,  and  the  contour  of  the 


DESCRIPTIVE   GEOMETRY. 


147 


sphere  in/*  and  K\  tlieuyA  istlie  horizontal  projection  of  the  ckcle 
cut  from  the  sphere,  and  its  middle  point  e  is  that  of  its  centre. 
Revolve  this  plane  about  ^  into  H;  o  goes  to  o'\  oo"  being  equal 
to  the  altitude  of  the  cone ;  and  do'\  ro'\  are  the  revolved  posi- 
tions of  the  elements  cut  from  the  cone.  Also,  e  goes  to  e'\  the 
distance  ee"  bemg  equal  to  that  of  C  from  H ;   about  e"  with  radius 


ef  describe  an  arc  cutting  do" ^  ro  ,  in  g 


u 


which  will  be  the 


-N  m 


revolved  positions  of  the  points  in  which  the  elements  pierce 
the  sphere.  In  the  counter-revolution,  g"  goes  to  g  on  pp,  whence 
it  is  vertically  projected  to  g'  on  d'o\  tims  determining  a  point  G 
on  tiie  required  curve ;  in  like  manner  the  projections  of  the  point 
whose  revolved  position  is  u"  may  be  found,  the  construction  being 
omitted  to  avoid  confusion  in  the  diagram. 

Otherwise:  Revolve  the  auxiliary  plane   about  the  horizontal 


148  DESCRIPTIVE   GEOMETRY. 

projecting  line  of  O  nntil  it  is  parallel  to  T ;  this  line  is  o'x  in  tlie 
vertical  projection,  and  setting  off  x'i\  =  or^  and  xd^  =  od^  the  two 
elements  appear  as  o'r^ ,  o'd^\  on  the  horizontal  through  c'  set  off 
ye^  =  oe^  then  e^  is  the  revolved  position  of  the  centre  about  which 
an  arc  with  radius  ^is  to  be  described,  cutting  the  elements  in  u^ 
and  (J,. 

221.  By  repeating  the  above  process,  any  number  of  points 
may  be  found  and  the  curve  fully  determined.  The  points  at  which 
the  projections  of  this  curve  are  tangent  to  the  extreme  visible  ele- 
ments of  the  cone,  as  for  instance  n'  on  o'm  ^  are  determined  as  in 
previous  cases,  by  passing  auxiliary  planes  through  those  elements. 
But  the  points  at  which  the  vertical  projection  is  tangent  to  the 
contour  of  the  sphere  cannot  be  located  by  any  direct  means,  since 
there  is  no  way  of  ascertaining  which  element  of  the  cone  will 
pierce  the  sphere  in  the  great  circle  cut  from  it  by  the  plane  II 
parallel  to  V.  These  points  may,  however,  be  determined  indi- 
rectly, thus :  the  horizontal  projection  of  the  curve  cuts  II  at  s  and 
w ;  these  are  the  horizontal  projections  of  those  points  of  penetra- 
tion, and  the  vertical  projections  s'  and  w'  must  lie  on  the  contour 
of  tlie  sphere. 

222.  In  this  instance,  the  vertex  of  the  cone  lies  within  the 
body  of  the  sphere ;  wdien  it  lies  outside  the  surface  there  may  be 
two  curves  of  intersection,  but  the  construction  is  the  same  for  both. 
If  any  plane  tangent  to  the  cone  passes  outside  the  sphere,  tlie  in- 
terpenetration  will  be  partial,  and  the  surfaces  will  intersect  in  one 
continuous  curve ;  if  the  cone  and  sphere  have  one  common  tan- 
gent plane,  this  curve  will  cross  itself  once ;  if  they  have  two  it 
will  cross  itself  twice.  In  order  to  ascertain  wdietlier  either  con- 
dition exists,  draw  a  line  from  the  centre  of  the  sphere  to  the  vertex 
of  the  cone,  and  make  it  the  axis  of  a  cone  of  revolution  with  the 
same  vertex  and  tangent  to  the  sphere ;  if  this  test  cone  is  tangent 
to"  the  given  one  along  one  element,  there  will  be  one  connnon  tan- 
gent plane ;  and  if  the  cones  are  tangent  along  two  elements,  there 
will  be  two  of  them.  If  the  cones  cut  each  other  in  two  elements, 
there  will  be  one  closed  curve  of  intersection ;  if  in  four  elements, 
the  given  cone  will  intersect  the  sphere  in  two  distinct  curves.  It 
is  here  assumed,  as  in  (212),  that  the  transverse  section  of  the 


DESCKIPTIVE   GEOMETRY.  149 

given  cone  by  a  plane  perpendicular  to  the  axis  of  tlie  test  cone,  is 
a  closed  curve  and  externally  convex  throughout. 

The  intersection  of  a  sphere  with  a  cylinder,  obviously,  is  to  be 
deter  mined  by  means  of  a  system  of  planes  cutting  elements  from 
the  latter  and  circles  from  the  former,  the  process  of  construction 
being  substantially  the  same  as  in  the  case  of  the  cone. 

223.  PiiOBLEM  3.    To  develop  any  ohlique  cone. 

Analysis.  Intersect  the  cone  by  a  sphere  whose  centre  is  at 
the  vertex.  The  curve  of  intersection  vrill  develop  into  an  arc  of 
a  circle  whose  radius  is  equal  to  that  of  the  sphere.  On  this  circle 
lay  oil'  distances  equal  to  the  rectiiied  arcs  of  the  curve  of  intersec- 
tion intercepted  between  rectilinear  elements  of  the  cone ;  and  on 
the  radii  drawn  through  the  points  thus  determined,  set  off  the  true 
lengths  of  the  corresponding  elements. 

Construction.  The  intersection  with  the  sphere,  in  Fig.  191,  is 
constnicted  as  in  Fig.  190.  Since  it  is  a  double  curved  line,  its 
true  length  is  not  seen  in  either  projection ;  but  it  can  be  ascer- 
tained by  developing  the  horizontal  projecting  cylinder,  as  shown  in 
Dj.  Here  the  elements  of  the  cylinder  appear  in  their  true  length 
and  at  their  true  distances  from  each  other ;  thus  ^,a?,  is  equal  to 
e'ic^  (j^y^  to  (j'y^  and  xij^  to  the  arc  eg  in  the  horizontal  projection ; 
the  positions  of  other  elements  being  determined  in  like  manner, 
the  base  develops  into  a  -right  line  w^w^.,  and  the  double  curve  line 
into  a  single  curved  one  nfijiy_ .  In  the  development  of  the  cone, 
shown  in  D^ ,  an  indefinite  circular  arc  is  described  about  any  centre 
6>2  with  a  radius  equal  to  that  of  the  sphere ;  on  this,  set  oif  an  arc 
e^(j^  equal  to  the  arc  e^g^  in  D^,  then  g^in^  =  ^,7?z, ,  and  so  on :  the 
points  thus  determined  fix  the  positions  of  the  elements  of  the  cone 
on  tlie  developed  sheet,  and  on  the  radii  drawn  through  them  the 
true  lengths  of  these  elements  are  set  off, — as  o.{Jl^  =^  OD^  o^f^  = 
0I'\  etc.  In  a  similar  manner  the  developed  form  of  any  other 
curve  on  the  surface  may  be  determined. 

To  draw  a  tangent  to  the  developed  base  at  any  point,  as  f^.  The 
line  o.f^  is  the  developed  position  of  the  element  OF^  and  the  tan- 
gent to  the  base  of  the  cone  at  F  is  FQ^  in  the  horizontal  plane. 
Find  the  true  angle  included  between  OF  and  FQ^  and  make  the 
angle  o^f^q^  equal  to  it;  theny!,^,  is  the  required  tangent.     And  the 


150 


DESCRIPTIVE   GEOMETRY. 


tangent  to  the  development  of  any  otlier  curve  on  the  surface  may 
be  drawn  in  the  same  way. 


Fig.  191 


224.  Conditions  of  Symmetry.  Tlie  cone  represented  in  Fig. 
191,  having  a  circular  base,  is  symmetrically  divided  by  tlie  verti- 
cal plane  through  tlie  axis,  which  also  cuts  from  it  the  longest  ele- 
ment OD  and  the  shortest  one  0C\  and  the  points  E^  N^  in  which 
these  elements  pierce  the  sphere,  are  respectively  the  highest  and 
the  lowest  points  of  the  curve  of  intersection.  Again,  any  two  ele- 
ments OF^  OK^  equidistant  from  OD^  will  be  of  equal  length, 
make  equal  angles  with  the  plane  of  the  base,  and  therefore  pierce 
the  sphere  in  points  of  equal  altitude ;  moreover,  since  they  are 
equally  foreshortened  in  the  horizontal  projection,  og  is  equal  to  ol, 
and  eg  equal  to  el- 


DESOKIPTIVE   GEOMETRY.  151 

From  these  considerations  it  follows  that  the  horizontal  projec- 
tion of  the  curve  of  intersection  is  symmetrical  with  reference  to 
od ;  that  the  development  of  the  projecting  cylinder  is  symmetrical 
about  e^x^ ;  and  that  the  development  of  the  cone  is  symmetrical 
about  o^d^ .  And  it  is  evident  that  the  same  will  hold  true  in  re- 
gard to  any  cone  which  is  symmetrically  divided  by  a  plane  through 
the  vertex  perpendicular  to  the  plane  of  the  base. 

225.  Practical  Suggestions.  The  above  deductions  are  of  im- 
portance in  the  practical  applications  of  this  problem.  For,  a 
curve  which  is  symmetrical  mth  respect  to  a  given  line  can  always 
be  constructed  more  easily  and  more  accurately  than  one  which  is 
not :  and  the  process  of  re-forming  the  original  cone  from  the  de- 
veloped sheet  is  likewise  much  facilitated  if  the  latter  be  symmetri- 
cal. 

Such  work  as  this  should  in  practice  always  be  laid  out  upon  as 
large  a  scale  as  may  be ;  when  this  is  done,  the  processes  of  rectifying 
the  arcs  of  the  horizontal  projection  of  the  curve  of  intersection,  in 
order  to  develop  the  projecting  cylinder,  and  of  transferring  the 
length  of  the  developed  curve  to  the  circular  arc  in  developing  the 
cone,  may  be  executed  with  sufficient  accuracy  by  stepping  them 
of[  with  the  spacing  dividers,  the  points  of  which  are  set  so  close 
together  that  the  difference  between  the  chord  and  the  arc  shall  be 
practically  inappreciable. 


152  DESCRIPTIVE   GEOMETRY. 


CHAPTER  YI. 

Of  Warped  Surfaces. 

The  Hyperbolic  Paraboloid ;  Its  Vertex,  Axis,  Principal  Diametric  Planes  and 
Gorge  Lines.  The  Conoid. — The  Hyperboloid  of  Revolution.— The  Ellipti- 
cal Hyperboloid,  and  its  Analogy  to  tlie  Hyperbolic  Paraboloid.  The 
Helicoid  ;  of  Uniform  and  of  Varying  Pitch.  The  Cylindroid. — The  Cow's 
Horn. — Warped  Surfaces  of  General  Forms. — Planes  Tangent  to  Warped 
Surfaces.  Warped  Surfaces  Tangent  to  Each  Other.  Interse3tions  of 
Warped  Surfaces. 

THE    HYPERBOIJC    PARABOLOID. 

226.  The  Hyperbolic  Paraboloid  is  a  warped  surface,  with  a 
plane  directer,  and  two  rectilinear  directrices  wliicli  lie  in  different 
planes.  It  takes  its  name  from  the  fact  that  its  curyed  sections  by 
planes  are  either  hyperbolas  or  parabolas. 

Any  plane  parallel  to  the  plane  directer  will  cut  each  directrix 
in  a  point;  and  the  right  line  joining  these  two  points  will  be  an, 
element  of  the  surface. 

If  a  series  of  such  parallel  planes  be  drawn,  they  will  divide  the 
two  directrices  proportionally.  And  conversely  :  If  any  two  right 
lines  not  in  the  same  plane  be  divided  into  proportional  parts,  the 
right  lines  joining  the  corresponding  points  of  division  will  lie  in 
parallel  planes,  and  be  elements  of  a  hyperbolic  paraboloid. 

If  through  any  point  in  space  two  lines  be  drawn,  respectively 
parallel  to  any  two  of  these  elements,  the  plane  of  those  two  lines 
will  be  parallel  to  the  plane  directer,  and  may  be  taken  for  it. 

If  in  this  j^lane  directer  any  right  line  be  drawn,  the  element 
parallel  to  that  line  may  be  found  thus :  Through  any  point  of 
either  directrix  draw  a  line  parallel  to  the  given  line ;  that  direc- 
trix and  this  parallel  will  determine  a  plane  cutting  the  other  direc- 
trix in  a  point :  through  that  point  draw  a  parallel  to  the  given 
line,  and  it  will  be  the  element  required. 


DESCRIPTIVE   GEOMETRY. 


153 


227.  The  Hyperbolic  Paraboloid  is  Doubly  Buled;  that  is  to 
say,  it  lias  two  sets  of  rectilinear  elements,  and  consequently  two 
plane  directers. 

In  Fig.  192,  let  XX  be  the  plane  directer,  A  O  smd  jBD  the 
directrices,  CD  and  AB  two  elements,  the  latter  lying  in  XX. 


Fig.  193 


Through  BD  draw  a  plane  T^T^  parallel  to  A  O,  cutting  XX  in 

TFTF:  a  plane  parallel  to  this  through  AC  cuts  XX  in  AZ  parallel 

to  WW,     Draw  Z>^ perpendicular  to  WW,  also  6^^ perpendicular 

to  AZ',  then  ^^will  be  parallel  and  equal  to  CD. 

Through  any  point  ^  on  ^  6^  draw  £[F  parallel  to  the  plane 

A  7^         T^Tf 
XX,  and  cutting  BD  in  F',  then  -^  =  -—  (226),  and  ^i^  will 

be  another  element  of  the  surface. 


154  DESCKIPTIYE   GEOMETRY. 

Draw  ^6^  perpendicular  to  J. Z  and  i^/ perpendicular  to  WW, 
then  GI  will  be  parallel  and  equal  to  £^2^.     We  have  also 

EG  -  GH'    ^^^    FD  ~  IK  ' 

therefore  j^fj^  =  y^,  consequently  JTIT  and  GI  cut  AB  in  the 

same  point  B. 

Now  draw  any  plane  parallel  to  YY,  cutting  AB  in  Z;  its 
intersection  with  JC^  will  be  parallel  to  AZ,  and  will  cut  GI  in 

Tvr    zxz^-      ir      •    •        ^^        ^^        ^^'       rpi-       1 
JV,  HK  m  jM,  giving  -y^  =  ttt/  =  ~Ttp'      ^^^^^  plane  also  cuts^ 

the  parallelogram  GF  in  iV^O  parallel  and  equal  to  GE^  and  the 
parallelogram  HD  in  MP  parallel  and  equal  to  IIC.     Therefore 

LN       NO      ^.,  ,       x^7.  .         .  1     T 

we  have    ■y^J^J^  =  jifl*'^    which  proves  that  LOF  is  a  riglit  line, 

intersecting  the  three  elements  AB,  FF,  CD.     Also,  the  elements- 

CD,  EF,  AB,  are  proportionally  divided  by  the  planes  parallel 

,.    CP       EG       AL 
to  YY,  so  that  -p-^  =  -Q^  =  z^- 

If  then  AB  and  CD  be  taken  as  directrices,  and  YY  as  a 
plane  directer,  the  resulting  surface  will  be  identical  with  that 
having  A  C  and  BD  as  directrices  and  JlX^  for  a  plane  directer. 
The  rectilinear  elements  CD,  EF,  etc.,  which  are  parallel  to  XX., 
are  called  elements  of  the  first  generation ;  those  of  the  other  set,  as- 
AC,  LP,  etc.,  which  are  parallel  to  YY,  are  called  elements  of 
tlie  second  generation.  And  from  the  preceding  it  appears  tliat 
every  element  of  either  generation  intersects  all  those  of  the  other. 

228.  Vertex  and  Axis.  It  is  clear  that  a  plane  through  R  par- 
allel to  YY,  in  Fig.  192,  will  determine  an  element  FT  of  the 
second  generation,  parallel  to  DK,  which  is  perpendicular  to  WW. 
In  a  similar  manner  (226)  an  element  of  the  first  generation  may 
be  found  which  shall  be  parallel  to  any  line  drawn  in  the  plane  XX 
and  perpendicular  to  WW.  Thus  in  Fig.  193,  having  found  RT 
as  above,  draw  AF  perpendicular  to  TFTF,  pass  through  C  a  plane 
parallel  to  XX,  cutting  J^I^in  DQ,  and  draw  CE,  perpendicular 
to  DQ  and  consequently  equal  and  j)arallel  to  AF.      Then  EFy 


DESCRIPTIVE   GEOMETRY. 


155 


equal  and  parallel  to  A  C\  cuts  BD  in  iT,  and  N'M  parallel  to  AF 
is  the  required  element. 

The  elements  BT^  MN^  therefore,  lie  in  a  plane,  which  is  per- 
pendicular to  TFTF,  the  intersection  of  the  two  plane  directers,  and 
cuts  it  at  /.  The  point  0  in  which  these  elements  cut  each  other 
is  called  tlie  vertex  of  the  surface ;  and  the  line  TJTI^  drawn  through 
0  parallel  to  TFTF,  is  called  the  axis.  The  plane  determined  by 
RT  and  MN^  again,  is  tangent  to  the  surface  at  the  vertex  0 
(142).   _  ■ 

Obviously,  the  angle  NOR  between  these  two  elements  is 
equal  to  the  angle  NIR  between  the  two  plane  directers :  if  this 
angle  is  a  right  angle,  the  surface  is  called  a  right,  or  rectangular, 
hyperbolic  paraboloid ;   if  not,  the  surface  is  called  oblique. 

229.  In  the  absence  of  a  model,  the  pictorial  representation  in 
Fig.  194  may  aid  in  forming  a  conception  of  this  surface.  NS  in 
the  horizontal  plane,  and  MT  in  the  vertical  plane,  are  divided 
into  the  same  number  of  equal  parts,  and  the  right  lines  joining  the 


Fig.  194 


corresponding  points  of  division  are  elements  of  one  generation; 
those  of  the  other  generation  are  determined  by  like  treatment  of 
>&l!fin  V  and  NT\u  H. 

Draw  Tt'  parallel  to  J/iS',  then  tTt'  is  the  plane  directer  of  the 
first  generation ;  in  like  manner  Ss'  parallel  to  MT  determines  s8s\ 


156 


DESCIlirTIVE    GEOMETRY. 


the  plane  directer  of  tlie  second  generation.  The  vertical  traces  of 
these  two  planes  intersect  at  G  on  the  vertical  plane,  their  hori- 
zontal traces  at  iV^on  the  horizontal  plane,  thus  determining  WW^ 
the  intersection  of  the  plane  directers. 

Since  N8  is  an  element  of  one  generation,  and  NT  one  of  the 
other,  the  horizontal  plane  is  tangent  to  the  surface  at  N  (142) ; 
and  for  a  like  reason  the  vertical  plane  is  tangent  to  it  at  M.  This 
is  most  clearly  shown  in  Fig.  195,  which  represents  the  same  sur- 
face in  projection  on  the  profile  plane. 

230.  The  Principal  Diametric  Planes.  Every  hyperbolic  parabo- 
loid, whether  rectangular  or  oblique,  is  symmetrically  divided  by 
two  planes  perpendicular  to  each  other,  passing  through  the  axis, 
and  bisecting  the  angles  formed  by  the  two  elements  which  deter- 
mine the  tangent  plane  at  the  vertex. 

Fig.  196  represents  in  profile  the  system  of  Fig.  193,  for  con- 


FiG.  198 


venience  so  placed  that  the  plane  RN  coincides  with  the  paper,  and 
that  the  plane  directers  are  equally  inclined  to  the  horizontal  plane 
HH,  wliich  contains  their  intersection  TFTF,  represented  by  the 


DESCRIPTIVE   GEOMETRY.  157 

point  /:  thus  the  axis  is  represented  by  the  point  (9,  and  the  ele- 
ments of  the  two  generations  by  the  Hnes  respectively  parallel  to 
XX  and  YT. 

Set  off  on  OB  the  points  a,  h,  and  on  0^  the  points  c,  d,  all 
equidistant  from  O ;  through  a  and  h  draw  parallels  to  XX,  and 
through  c  and  d,  parallels  to  YY:  the  intersections  of  these  paral- 
lels determine  the  rhombus  egfh.  Through  e  and  f  draw  the  plane 
ZZ,  and  through  g  and  k  the  plane  i^P,  both  perpendicular  to  tlie 
paper;  these  planes  are  perpendicular  to  eacli  other,  intersect  in  the 
axis  0,  and  bisect  the  angles  formed  by  Oli  and  OJV^. 

Now,  the  sides  of  the  rhombus  represent  four  elements  of  the 
surface;  and  since  a,  h,  c,  d,  lie  in  the  plane  of  the  paper,  the 
intersections  e,  y,  g,  h,  do  not,  because  those  elements  are  inclined 
to  that  plane.  Suppose  e  to  lie  at  any  distance  behind  it;  then 
because  elj,  hh,  hd,  d/,  etc.,  are  all  equal,  the  point /*  must  also  lie 
at  tlie  same  distance  beliind  it ;  and  the  points  g,  A,  at  exactly  the 
same  distance  in  front  of  it. 

Set  off  hk  on  IcO  equal  to  cs  on  JV^O,  and  complete  the  rhom- 
bus mpnq,  then  by  the  same  reasoning  it  appears  that  m  and  n  are 
at  the  same  distance  behind  the  plane  of  the  paper,  and  p,  q,  at  an 
equal  distance  in  front  of  it.  Produce  the  sides  of  the  lirst  rhombus 
to  cut  those  of  the  otlier,  then  r,  I,  y,  w^  are  at  equal  distances 
beliind  the  plane,  while  lo,  t,  v,  a?,  are  at  equal  distances  in  front 
of  it.  Then  rl,  sk,  ut,  etc. ,  are  chords  of  the  surface  parallel  to 
the  planes  PP  and  PX;  and  the  plane  ZZ  bisects  them  all.  So 
also  all  chords  parallel  to  ZZ  and  7?X,  as  for  example  tv,  ly,  mn, 
etc.,  are  bisected  by  PP.  Consequently,  as  stated,  the  surface  is 
symmetrical  with  respect  to  both  planes;  which  are  called  the 
principal  diametric  planes. 

231.  Tlie  Gor^e  Lines.  There  may  be  an  indefinite  number  of 
hyperbolic  paraboloids  having  the  same  plane  directors  XX,  YY, 
and  the  profile,  Fig.  196,  represents  them  all.  JS'ow,  the  point  e 
may  be  at  any  distance  beyond  the  plane  of  the  paper ;  but  if  that 
distance  be  assigned,  the  inclination  of  the  directrices  to  that  plane 
is  thereby  fixed,  and  the  surface  fully  determined. 

Let  Ok  be  twice  Oh ;  then  since  el  —  ce,  I  will  be  twice  as  far 
beyond  the  plane  as  e,  and  since  Im  —  Ik,  m  will  be  twice  as  far 


158  DESCRIPTIVE    GEOMETRY. 

beyond  it  as  Z,  or  four  times  as  far  as  e ;  wliile  the  distance  of  ra 
from  the  axis  is  but  twice  that  of  e. 

Had  Oh  been  three  times  Oh^  m  would  have  been  three  times 
as  far  as  e  from  the  axis,  but  nine  times  as  far  beyond  the  plane  of 
the  paper ;   and  so  on. 

Consequently  LL  cuts  the  paraboloid  in  a  curve,  symmetrical 
with  respect  to  the  axis  of  the  surface,  having  its  vertex  at  0^  ex- 
tending to  infinity  beyond  the  plane  i?^^,  and  having  its  abscissas 
proportional  to  the  squares  of  the  ordinates ;  that  is  to  say,  in  a 
parabola.  And  in  a  similar  manner  it  may  be  shown  that  PP 
cuts  the  surface  in  another  parabola  having  its  vertex  at  0,  and  ex- 
tending in  front  of  the  plane  PI^. 

232.  Since  the  plane  directers  in  Fig.  196  are  equally  inclined 
to  the  horizontal  plane,  PP  is  horizontal  and  LL  is  vertical.  Fig. 
197  is  a  projection  upon  the  latter  plane;  in  which  the  planes  PP 
and  PN  are  seen  edgewise.  Set  off  01 ,  the  assumed  distance  of 
e  from  PN^  and  Og  equal  to  it ;  also  set  off  6^4  and  Op  equal  to 
four  times  01.  Taking  the  lengths  of  the-  chords  ef^  mn^  from 
Fig.  196,  draw  the  projections  of  the  rectilinear  elements  through 
9")  ^j  ^j/*?  i^?  ^^?  andj9,  n.  Thus  the  parabola  mOn  is  shown  in 
its  own  plane ;  also,  it  is  seen  that  ge^  j^m,  are  tangent  to  this  curve 
at  e  and  m,  (the  subtangents  l)eing  bisected  at  the  vertex) :  which 
is  as  it  should  be,  because  eg.,  <?A,  being  elements  of  different  gener- 
ations, deteiTuine  a  plane  tangent  to  the  surface  at  their  intersec- 
tion e  (142).  The  other  parabola  is  shown  in  its  true  form  in  Fig. 
198,  which  is  a  projection  on  the  plane  PP:,  the  construction  is 
obvious,  the  lengths  of  the  chords  gh^  jpq^  being  transferred  from 
Fig.  196. 

Tliese  parabolas  are  upon  different  scales,  since  the  chords  ef^ 
gh^  equidistant  from  the  vertex,  are  respectively  equal  to  the  diago- 
nals of  a  rhombus;  this  is  because  this  particular  surface  is  an 
oblique  one :  liad  it  been  rectangular,  the  rhombus  would  evidently 
have  been  a  square,  its  diagonals  equal,  and  the  parabolas  identical. 

233.  If  in  Fig.  197  a  plane  parallel  to  PNhe  drawn  through 
any  point  as  ^,  it  will  cut  any  pair  of  elements  abov^e  6,  Sispm^  qm, 
thus  determining  a  chord  of  the  curve  of  intersection,  seen  as  P£! 
in  Fig.  198.     By  drawing   other  elements,  other  points    of   tliis 


DESCRIPTIVE   GEOMETRY.  159 

<iurve  are  easily  found ;  and  it  is  clear  that  e  must  be  its  vertex, 
])ecause  the  elements  (je,,  he^  determine  a  plane  tangent  to  the  sur- 
face at  that  point,  and  that  plane  intersects  the  cutting  plane  in  a 
line  perpendicular  to  LL,  which  is  the  tangent  to  the  curve  at  the 
point  under  consideration.  The  same  plane  will  cut  the  lower 
part  of  the  surface  in  a  similar  and  opposite  curve,  as  shown  in 
Fig.  199,  which  is  a  projection  on  the  profile  plane  BW.  In  this 
figure  (7,  C^  are  the  curves  cut  from  the  surface  by  the  plane 
tin-ough  e^  and  2>,  Z>,  are  tliose  cut  from  it  by  a  parallel  plane 
through  ^,  at  the  same  distance  from  EN  but  on  the  opposite  side. 
It  can  be  proved  tliat  these  curves  are,  in  fact,  hyperbolas,  of 
which  the  centres  lie  in  the  axis  of  the  surface,  and  the  asymptotes 
are  parallel  to  the  elements  Oli^  0N\  but  without  discussing  their 
mathematical  peculiarities,  it  is  evident  from  what  precedes  that 
they  are  convex  toward  the  axis,  and  that  their  vertices  lie  upon 
the  two  parabolas  shown  in  Figs.  197  and  198:  which  suffices  to 
show  that  those  parabolas  are  true  gorge  lines,  as  defined  in  (129). 

234.  The  Plane  a  Limiting  Form.  Referring  to  Fig.  194,  sup- 
pose the  elements  to  be  perfectly  elastic  lines,  fixed  at  each  end  in 
the  two  planes  H  and  V.  Then  if  V  be  revolved  about  the  ground 
line  in  the  direction  of  the  arrow,  these  elements  will  be  length- 
ened, the  curvature  of  the  surface  becoming  less  and  less,  until  at 
the  limit,  when  J!/^  falls  in  H  beyond  the  ground  line,  the  parabo- 
loid will  have  been  extended  into  a  plane. 

If  on  the  other  hand  V  be  revolved  in  the  opposite  direction, 
the  curvature  of  the  surface  will  become  greater  and  greater,  the 
elements  contracting,  until  Jf  falls  in  H  in  front  of  the  ground  line, 
and  the  paraboloid  will  have  collapsed  into  a  plane. 

235.  The  Warped  Quadrilateral.  Four  right  lines,  connecting 
any  four  points  not  in  the  same  plane,  constitute  what  is  sometimes 
<?alled  a  loarjped  qtiadr Hater al^  and  may  define  a  portion  of  a 
hyperbolic  paraboloid  :  not  necessarily,  however,  since  there  are 
other  warped  surfaces  which  are  doubly  ruled,  and  such  a  quadri- 
lateral can  evidently  be  drawn  upon  any  one  of  them. 

But  supposing  that  they  do ;  it  is  to  be  remarked  that  the  same 
four  points  may  be  joined  two  and  two 'by  right  lines  in  three  dif- 
ferent ways,  thus  forming  three  different  quadrilaterals,  as  shown 


160 


DESCKTPTIVE    GEOMETRY. 


ill  Figs.  200,  201,  and    202,   and  determining   as   many  diverse 
hyperbolic  paraboloids. 


Fig.  201 


Tig.  202 


Fig.  200 

236.  Representation  of  the  Surface.  In  Fig.  203,  draw  ^Z^^ 
MN^  any  two  right  lines  of  definite  length  not  lying  in  the  same 
plane,  and  divide  each  into  the  same  number  of  equal  parts ;  the 
lines  joining  the  corresponding  points  of  division  will  be  elements 
of  a  hyperbolic  paraboloid  (226). 


i>' 

r 

'^" 

— 

^ 

:^ 

-/ 

^'/ 

o 

^ 

rr- 

/ 

/ 

_. 

-4' 

tA 

7^ 

h' 

Fig.  203 


Fig.  204 


To  assume  a  point  upon  this  surface.  Assume  the  horizontal 
projection,  as  6>,  Fig.  203 ;  the  point  must  then  lie  upon  a  vertical 
line  through  o\  of  which  h't'  is  the  vertical  projection.  Through 
this  vertical  line  pass  any  plane,  as  II ;  it  cuts  the  elements  at  Gy 
E^  etc.,  thus  determining  a  curVe  of  intersection,  whose  vertical 
projection  g'k'It  cuts  h't'  in  o\  the  vertical  projection  of  the  as- 
sumed point :  and  in  a  similar  manner  the  horizontal  projection 
may  be  found  if  the  vertical  projection  is  assumed.     In  either  case 


DESCRIPTIVE    GEOMETRY.  161 

tlieiv^  maj  be  two  determinations,  since  a  right  line  may  pierce  the 
surface  in  two  points ;  but  no  more. 

The* same  process,  obviously,  is  applicable  to  any  otlier  warped 
eurface. 

To  draw  an  element  through  the  point  thus  found.  In  Fig.  204, 
the  quadrilateral  P 31  and  the  point  O,  in  order  to  avoid  confusion 
in  the  diagram,  are  cojDied  from  Fig.  203.,  Through  0  draw  OB 
parallel  to  MJV,  and  O/S  parallel  to  Pi>;  lind  as  in  Fig.  83  the 
points  X  and  Y,  in  which  the  plane  of  these  two  lines  cuts  DM 
and  PJV:  then  XO  Y  is  an  element  of  the  surface. 

To  draw  a  plane  tangent  to  the  surface  at  this  point.  Draw 
through  O  a  plane  parallel  to  Z>i¥  and  PiY,  and  find  as  above  the 
points  in  which  it  cuts  PD  and  MN-^  then  WOZ  drawn  through 
those  points  is  an  element  of  the  other  generation,  and  the  plane 
determined  by  XY  oiid  WZ  is  tangent  to  the  surface  at  O  (142) : 
and  its  traces  if  required  may  be  found  in  the  usual  manner. 

237.  Through  any  two  given  right  lines  not' in  the  same  plane, 
any  number  of  hyperbolic  paraboloids  may  be  passed ;  for  taking 
those  two  lines  as  directrices,  there  may  be  an  infinite  number  of 
plane  directers.  This  may  be  seen  from  another  point  of  view,  by 
consider ing  that  the  ratio  between  the  lengths  of  PP  and  31 X^  in 
Fig.  203,  is  entirely  arbitrary ;  and  that  any  change  in  this  ratio 
must  modify  the  form  of  the  surface,  since  one  of  the  plane  direct- 
ers is  always  parallel  to  DM  and  PX^  and  the  other  to  PD  and 
3IX,  It  follows  from  this  that  if  any  two  right  lines  not  in  the 
same  plane  be  drawn  of  indefinite  length,  and  equal  spaces  be  set  off 
upon  each  of  them  by  any  two  scales  of  equal  parts,  whatever  the 
ratio  between  the  units  of  those  scales :  the  right  lines  joining  the 
successive  points  of  division  will  be  elements  of  a  hyperbolic  para- 
boloid. 

238.  Practical  Applications.  The  hyperbolic  paraboloid  is 
sometimes  met  with,  forming  the  basis  if  not  the  actual  surface  of 
a  practical  structure,  without  revealing  its  true  nature  to  the  casual 
observer.  Thus  in  Fig.  205,  A  is  a  front  view  and  B  is  a  top  view 
of  the  "pilot,"  or  "cow-catcher,"  of  an  American  locomotive, 
drawn  of  course  in  skeleton  ;  tins  being  divided  by  the  central  ver- 
tical plane  LL^  C  is  a  view  from  one  side,  and  D  a  view  from  the 


16'^ 


DESCRIPTIVE    GEOMETRY. 


otiier  side,  of  that  plane.  Considering  first  the  part  on  tlie  left  of 
LL  in  view  A,  and  regarding  the  warped  quadrilateral  aoeb  as  lying 
upon  a  surface  of  tliis  kind;  then  by  subdividing  ao.,  he^  as  above 
explained,  we  determine  a  series  of  elements  parallel  to  LL^  as 


r   L 


Fig.  205 


a 

A 

L 

0 

e 

C 

h 

r 

L 

shown  in  view  C  and  in  the  lower  half  of  view  B.  Upon  the  other 
side  of  LL  there  is  a  warped  quadrilateral  scrd^  symmetrical  to  the 
first ;  in  this  case,  by  means  of  corresponding  subdivisions  of  cr  and 
scl^  a  series  of  horizontal  elements  is  determined,  as  shown  in  view 
D  and  in  the  upper  half  of  view  B. 

The  pilot,  then,  is  composed  of  two  symmetrically  placed  hy- 
perbolic paraboloids,  of  which  the  horizontal  plane  and  the  vertical 
plane  LL  are  the  common  plane  directers ;  and  the  bars  may  lie 
either  in  vertical  planes  or  in  horizontal  planes :  both  arrangements 
have  been  used,  the  preference  being  given  to  the  former. 

239.  This  surface  has  also  been  employed,  in  designing  the  bow 
of  a  boat,  as  shown  in  Fig.  206,  where  the  water-lines  are  shown  at 
A,  while  B  is  the  sheer  plan  and  C  the  body  plan.  The  warped 
quadrilateral  dhco  being  treated  as  in  Fig.  200,  the  lines  xx^  etc.,  as 
well  as  the  frames  nn,  etc. ,  are  right  lines,  as  shown  not  only  in  B 
but  in  the  left-liand  part  of  C  and  the  lower  half  of  A.  But  the 
true  water-lines,  1,  2,  3  (or  horizontal  sections,  as  shown  in  the 
right-hand  half  of  C),  are  not  straight,  but  what  is  technically  called 
liollow, — that  is,  outwardly  concave,  as  seen  in  the  upper  half  of  A. 
This  circumstance  is  obviously  due  to  tlie  fact  tliat  dh  in  this  case  is 


DESCRIPTIVE   GEOMETRY. 


Xt)3 


not  horizontal,  but  droops  as  it  recedes  from  do^  the  vertical  line  of 
the  stem ;  which  is  most  clearly  shown  in  the  sheer  plan  B.     And 


'      h 

y 

so 


i> 


- 

p 

m 

n 

y 

\-  —4 

X 

J\ 

p  m  n 


Fig,  206 


it  is  equally  clear  that  if  dh  had  been  horizontal,  all  the  water-lines 
would  have  been  straie^lit  instead  of  concave. 


THE    CONOID. 

240.  The  Conoid  is  a  warped  surface,  with  a  plane  directer, 
and  two  linear  directrices,  one  of  which  is  a  right  line,  and  the 
other  a  curve.  Thus  the  elements  of  the  surface,  instead  of  con- 
verging  to  a  single  point  as  in  the  cone,  pass  through  all  the  points 
of  the  rectilinear  directrix.  If  that  directrix  is  perpendicular  to  the 
plane  directer,  it  is  called  the  Line  of  Striction,  and  the  surface  is 
called  a  Right  Conoid. 

The  definition  being  perfectly  general,  the  curved  directrix  may 
be  of  single  or  of  double  curvature.  But  the  term  conoid  is  fre- 
quently used  in  a  limited  sense,  as  including  only  those  surfaces  in 
which  this  directrix  is  a  closed  curve,  lying  in  a  plane  perpendicular 
to  the  plane  directer. 

241.  In  Fig.  207  is  represented  the  most  simple  of  the  conoids, 
which  bears  to  others  the  same  relation  that  the  cone  of  revolution 
has  to  all  other  cones ;  and  in  all  probability  gave  the  name  to  the 
class.     This  is  a  right  conoid,  of  which  the  plane  directer  is  V,  and 


164 


DESCRIPTIVE    G  p]OM  ETRY. 


the  curved  directrix  a  circle  lying  in  H ;  the  line  of  striction  inter- 
sects at  O  the  axis  of  this  circle,  which  is  evidently  a  line  of  sym- 
metry, and  may  be  called  the  axis  of  the  surface.  It  is  apparent 
that  this  surface  is  symmetrically  divided  by  two  planes  through  the 
axis,  one  of  which  is  parallel,  and  the  other  perpendicular,  to  the 
plane  directer;  also  that  it  is  divided  by  the  line  of  striction  into 


'\ 

Ji 

7i, 

'\ 

n              \ 

\\fc'        \^ 

/ 

0, 

d'//  7 ' 

n<     ^ 

.  /  '' 

e'            Z' 

XV  ^' 

Ci 

r 

V. 

^.  / 

/w 

/  /        I 

\ 

\  A 

f    1         I 

I    d     g 

0      Ic 

/  c 

/■ 

Pig.  207* 

w\      p 

a'i 

'    z/  / 

u     ' 

•  \     \ 

./> 

/ 

T 

T                     ^**"-^^-»i 

^^■7/ 

^           ^m 

7/ 

7 

two  nappes,  like  a  cone,  which  are  similar  and  symmetrically  placed. 

The  intersection  of  this  conoid  by  a  plane  perpendicular  to  the 
axis  is  an  ellipse. 

Thus,  let  ZZ  be  such  a  i3lane,  lying  between  the  directrix  and 
the  line  of  striction :  then  it  is  seen  that  in  the  horizontal  projec- 
tion the  ordinates  go,  rx,  are  less  than,  but  directly  proportional 
to,  the  corresponding  ordinates  do,  yx,  of  the  circle ;  therefore  the 
curve  is  an  ellipse  whose  major  axis  mn  is  equal  to  the  diameter  of 
the  circle.  If  the  elements  be  extended,  -and  cut  by  a  horizontal 
plane,  as  II,  more  remote  than  the  directrix  from  the  line  of  stric- 


DESCRIPTIVE    GEOMETRY.  165 

tion,  the  ordinates  will  be  greater  than  those  of  the  circle,  and  the 
minor  axis  of  the  elliptical  section  shown  in  a  dotted  line  will  be 
equal  to  the  diameter  of  the  directrix. 

242.  Planes  Tangent  all  along  Rectilinear  Elements.  It  is  verj 
*easy  to  draw  the  plane  tangent  at  any  point,  as  A  for  instance, 

since  it  must  contain  the  element  through  that  point,  and  also  the 
tangent  at  the  same  point  to  the  elliptical  section  by  a  horizontal 
plane;  the  horizontal  trace  of  the  required  plane  being  drawn 
through  ^,  the  foot  of  the  element,  and  parallel  to  that  taiigent,  the 
vertical  trace  is  drawn  parallel  to  az' .  And  it  is  obvious  that  in 
general  such  a  plane  will  not  be  tangent  along  the  element,  because 
the  tangent  to  the  circle  at  z  is  not  parallel  to  the  tangent  to  the 
ellipse  at  a. 

But  the  plane  TT^  parallel  to  V,  contains  the  vertical  element 
EH^  and  also  a  tangent  at  one  of  the  vertices  to  every  elliptical 
section :  it  is  therefore  tangent  all  along  the  element. 

So,  again,  the  plane  sSs\  perpendicular  to  T  and  containing 
CP^  is  tangent  to  the  surface  all  along  that  element ;  and  two  other 
planes  possessing  this  peculiarity  can  be  drawn  on  the  opposite 
sides. 

243.  In  Fig.  208,  Y  is  the  plane  directer,  the  curved  directrix 
is  the  horizontal  circle  CC^  and  the  rectilinear  directrix  DD  is  hori- 
zontal but  inclined  to  T,  and  intersects  at  O  the  axis  of  the  circle. 
The  upper  nappe  of  the  surface  is  not  represented,  but  the  elements 
are  continued  below  CC  to  the  horizontal  plane. 

It  is  obvious  on  inspection  that  this  surface  is  not  symmetrical 
like  the  preceding  one ;  the  two  nappes  will  not  be  similar,  and  the 
directrix  is  the  only  circular  section. 

In  regard  to  the  horizontal  trace,  we  have,  since  the  elements 
are  proportionally  divided  by  parallel  planes,  the  values 


fg       sv  _0G  _ 

an      de 

gk  ~  vw      Gu 

m      em ' 

therefore  that  trace  is  an  ellipse,  and  the  same  is  true  of  a  section 
by  any  horizontal  plane  as  LL. 

244.   Since  there  must  be  a  vertical  element  at  R  and  another 
at  S^  it  follows  that  rs  will  be  a  common  diameter  of  aU  these 


166 


DESCRIPTIVE    GEOMETRY. 


ellipses,  as  well  as  of  tlie  circle.  Draw  gn^  a  diameter  of  the  cir- 
cle perpendicular  to  rs^  and  through  g  and  n  draw  the  projections 
of  two  elements,  determining  the  points  /»•  and  z  in  the  horizontal 
trace.      Then  regarding  rs  as  a  diameter  of  the  ellipse,  kz  will  be 


the  conjugate  diameter ;  and  in  like  manner,  in  any  other  horizon- 
tal section,  the  extremities  of  the  diameter  conjugate  to  rs  will  lie 
xv^onfgk  and  anz.  The  tangents  to  these  ellipses  at  the  extremi- 
ties of  these  conjugate  diameters  are  parallel  to  rs^  consequently 
the  planes  determined  by  DD  and  the  eleinents  FK^  AZ,  are  tan- 


DESCRIPTIVE    GEOMETRY.  167 

gent  to  the  surface  all  along  those  elements ;  the  true  angles  y5,  &?, 
which  these  planes  make  with  H,  are  seen  in  the  supplementary 
projection. 

Draw  (?<?,  ex^  two  conjugate  diameters  of  the  directrix,  respec- 
tively parallel  and  perpendicular  to  V.  The  elements  through  the 
extremities  of  the  first  determine  the  points  u^  y^  of  the  horizontal 
trace ;  then  reasoning  as  before,  m^  the  extremity  of  the  diameter 
conjugate  to  ^ty^  must  lie  upon  the  element  through  e :  so  also  does 
j9,  the  extremity  of  a  diameter  conjugate  to  hi^  in  the  section  by 
the  plane  LL.  From  which  it  appears  that,  in  this  case  also,  two 
planes  parallel  to  V  can  be  drawn,  each  tangent  all  along  an  ele- 
ment. 

245.  Limiting  Forms  of  the  Conoid.  These  two  examples  illus- 
trate sufficiently  for  our  j^urpose  the  peculiarities  of  this  surface, 
which,  however,  is  susceptible  of  variations  without  number,  since 
the  straight  directrix  may  have  any  position,  and  the  curved  one 
any  form ;  in  consequence  of  this,  it  happens  that,  in  special  cases, 
warped  surfaces  of  other  classes  are  also  in  fact  conoids. 

The  less  the  inclination  of  the  straight  directrix  to  the  plane 
director,  the  more  acute  will  be  the  angles  between  it  and  the  ele- 
ments ;  and  at  the  limit,  when  that  directrix  is  parallel  to  the  plane 
director,  the  elements  will  become  parallel  to  it  and  to  each  other. 
In  that  event,  if  the  other  directrix  be  of  double  curvature,  or  lie 
in  a  plane  which  cuts  the  rectilinear  directrix,  the  conoid  will  be- 
come a  cylinder ;  if  it  lie  in  a  plane  parallel  to  the  rectilinear  direc- 
trix, the  surface  will  degenerate  into  a  plane. 

THE    HYPERBOLOID    OF    REVOLUTION. 

246.  This  surface  is  generated  by  a  right  line,  Avhich  revolves 
about  an  axis  lying  in  a  different  plane.  In  Fig.  ^09,  let  the  axis 
be  vertical,  the  point  o  being  its  horizontal,  and  w'o'y  its  vertical 
projection ;  and  let  the  generatrix  MN  be  parallel  to  Y  :  then  each 
point  of  MN  will  describe  a  horizontal  circle,  of  which  the  true 
radius  is  seen  in  the  horizontal  projection,  and  the  exact  altitude  in 
the  vertical  projection. 

Thus,  tlie  point  N  describes  a  circle  whose  radius  is  on ;  the 
vertical  projection  of  this  circle  is  determined  by  setting  off  on  each 


168 


DESCRIPTIVE    GEOMETRY. 


side  of  the  axis,  on  tlie  horizontal  through  n\  the  distance 
y'h'  =  on :  similarly,  the  vertical  projection  of  the  circle  described 
by  the  point  Ehfe'f\  where  u'f  ~  oe,  and  so  on.  The  circle  of 
least  diameter  is  that  described  by  the  point  O,  since  oc  is  the  com- 
mon  perpendicular  of  the  axis  and  the  generatrix.     Tiiis  circle, 


TiQ.2.09  FiQ.  210 

whose  vertical  projection  is  d'c'd' ^  is  called  the  gorge  circle,  and  its 
plane  is  called  the  gorge  plane.  Since  points  of  JfiT which  are  equi- 
distant from  C  are  also  equally  distant  from  the  axis,  and  therefore 
describe  equal  circles,  it  follows  that  the  portion  d'V  of  the  contour 
above  d'd'  is  precisely  like  the  portion  d'h'  below  that  line ;  and 
that  the  surface  is  symmetrically  divided  by  the  plane  of  the  gorge. 
247.  Practical  Suggestions.  This  surface  has  practical  applica- 
tions in  mechanism ;  and  it  is  proper  here  to  point  out  most  em- 
phatically, tliat  in  constructing  the  contour  it  is  not  only  useless, 
but  worse  than  useless,  actually  to  draw  the  vertical  projections 
d'd',  ff\eiG.^  thus  covering  the  paper  with  superfluous  lines  to  be 
subsequently  erased :  no  good  draughtsman  will  do  it,  nor  w^ill  any 


DESCRIPTIVE    GEOMETRY.  1C9 

good  instructor  permit  pupils  to  adopt  such  slovenly  methods.  It 
is  not  even  necessary  to  draw  the  horizontal  projections  of  the  radii ; 
having  marked  the  projections  of  any  point  of  J/iV,  as  r,  r'-^  mark 
on  the  axis  the  point  x  on  the  horizontal  through  r\  then  taking 
the  distance  or  in  the  compasses,  a  very  short  arc  only  is  drawn  as 
at  g\  g\  on  each  side,  with  x'  as  a  centre ;  the  precise  points  on 
these  arcs  being  finally  marked  on  the  horizontal  through  z' . 

248.  This  Surface  is  Doubly  Ruled.  The  portion  here  repre- 
sented is  limited  by  two  planes  equidistant  from  the  gorge  plane; 
which  may  be  called  its  bases.  Projecting  m  to  the  lower  base  at 
s\  and  n  to  the  upper  base  at^',  it  is  seen  that  Tucn  is  the  horizontal 
projection  of  a  second  line  whose  vertical  projection  is  s'g'2>' -  This 
line  makes  the  same  angle  as  MN^  Avith  the  plane  of  the  base,  but 
elopes  in  tlie  opposite  direction ;  moreover,  it  is  also  a  generatrix  of 
the  surface,  because  each  of  its  points,  in  revolving  around  the  axis, 
describes  the  same  circle  as  a  corresj)onding  point  of  MN\  for  in- 
stance, Z  and  R^  on  the  same  side  of  the  gorge  and  equally  distant 
from  it,  are  also  equidistant  from  the  axis,  and  describe  the  same 
horizontal  circle.  This  facilitates  the  construction  of  the  contour : 
thus  a'  is  vertically  over  e\  consequently  those  two  points  are  equi- 
distant from  {?',  and  the  circles  drawn  through  them  will  have  the 
same  radius  oe ;  the  radius  of  the  circle  through  V ^  vertically  over 
r%  is  equal  to  or^  and  so  on. 

249.  Through  any  point  of  the  surface,  then,  two  elements  can 
be  drawn ;  and  it  is  apparent  that  either  of  them  if  produced  will 
intersect  all  those  of  the  other  generation  except  that  one  which  is 
parallel  to  it,  for  the  simple  reason  that  they  do  lie  upon  the  same 
surface ;  and  it  may  be  said  to  intersect  that  one  at  an  infinite  dis- 
tance. From  this  it  follows  that  if  any  three  elements  of  either 
generation  be  taken  as  directrices,  any  element  of  the  other  may  be 
taken  as  a  generatrix,  whose  motion  will  produce  the  same  surface. 
Considered  in  this  light,  the  surface  is  one  with  three  rectilineat 
directrices ;  regarded  as  a  surface  of  revolution,  it  is  one  with  a 
cone  directer  and  two  circular  directrices, — the  former  being  a  right 
circular  cone  whose  ano-le  at  the  vertex  is  s'c'n'  in  FIh".  209 :  the 
base  of  which  is  II  in  the  horizontal  projection :  whan  thus  situ' 
ated  this  cone  is  obviously  asymptotic  to  the  surface. 


170  DESCRIPTIVE   GEOMETRY. 

If  a  plane  be  drawn  cutting  tliis  cone  directer  in  an  ellipse,  a 
parabola,  or  an  hyperbola,  a  plane  j^arallel  thereto  will  cut  this 
hyperboloid  in  a  curve  of  the  same  kind. 

250.  To  assume  a  point  on  the  surface.  In  Fig.  210,  the  contour 
having  been  found  as  above,  assume  the  vertical  projection  c  ;  then 
a  horizontal  through  c'  determines  the  radius  y'x'  of  the  circle  upon 
whose  circumference  c  must  lie.  Conversely,  if  the  horizontal  pro- 
jection c  be  assumed;  a  circle  through  c,  about  centre  <?,  will  be  cut 
by  the  meridian  plane  LL  parallel  to  V,  in  the  point  a?;  which  pro- 
jected to  the  contour  at  x'  will  determine  the  altitude  of  the  vertical 
projection  c' . 

Otherwise^  if  the  horizontal  projection  c  be  assumed :  Draw 
through  c  two  tangents  to  the  gorge  circle ;  these  are  the  horizontal 
projections  of  the  two  elements  which  pass  through  the  assumed 
point.  One  of  these  pierces  the  lower  base  in  D  and  the  gorge 
plane  in  i^;  the  other  pierces  the  lower  base  in  E  and  the  gorge 
plane  in  U\  and  their  vertical  projections  (^y,  eic\  intersect  in  c\ 
the  vertical  projection  of  the  assumed  point. 

To  draw  a  plane  tangent  to  the  surface  at  any  point.  Draw  through 
the  point  an  element  of  each  generation.  Thus  in  Fig.  209,  the 
two  elements  whose  horizontal  projection  is  ran  determine  the 
plane  tangent  to  the  surface  at  the  point  C  on  the  gorge  circle. 
And  in  Fig.  210  the  \?lements  DF^  EU^  determine  the  plane  tan- 
gent at  the  point  C\  in  this  figure  the  lower  base  coincides  with  H, 
consequently  tt^  drawn  through  d  and  e^  is  the  horizontal  trace  of 
the  plane.  This  trace  is  perpendicular  to  the  radius  oc^  which  fact 
enables  us  to  draw  it  with  precision,  even  when  d  and  e  are  very 
close  together ;  and  indeed  it  enables  us  to  dispense  altogether  with 
one  of  the  r.Iements  through  C^  since,  the  direction  being  known, 
either  one  of  the  points,  d  or  e^  suffices  to  locate  the  trace. 

251.  It  is  to  be  observed  that,  having  determined  the  circular 
paths  of  any  two  points  in  the  generatrix,  as  for  instance  Jf  and  N 
in  F^.g.  209,  the  projections  of  any  number  of  elements  might  have 
been  drawn ;  and  instead  of  constructing  the  contour  line  h'd'V  by 
points  as  above,  we  could  then  have  drawn  it  as  the  enveloj^e  of  the 
vertical  projections  of  those  elements  (170),  each  of  wliich  would  be 
tangent  to  it  at  a  point  in  the  meridian  plane  parallel  to  V :  thus  in 


DESCKIPTIVE    GEOMETRY.  171 

Fig.  21C,  tlie  element  EG  lias  but  one  point,  K^  in  common  with 
the  meridian  curve,  and  no  point  either  of  the  element  or  the  sur- 
face lies  to  the  left  of  that  curve ;  consequently  h'  is  a  point  of  tan- 
gency  in  the  vertical  projection. 

252.  The  Meridf.a.'»  Curve  is  an  Hyperbola.  In  Fig.  209,  the  in- 
tercepts e'f'^  r'g\  etc.,  diminish  as  they  recede  from  the  gorge. 
In  fact,  each  intercept,  as  r^g'  for  instance,  is  the  difference  between 
the  hypothenuse  or  and  the  base  cr  of  a  right-angled  triangle 
whose  altitude  oe  remains  constant.  Let  the  hypothenuse  =  h,  the 
base  —  b,  the    altitude  =  a;  then  a'  z=  h'  —  b'  =:  (h  -|-  b)(h  —  b), 

JS^ow  the  value  of  this  fraction  diminishes  as  the 


h-f-  b 

denominator  increases,  and  reduces  to  zero  when  h  +  b  becomes  in- 
finite :  consequently  c'n\  c'jp\  are  asymj)totes  to  the  meridian  curve 
l:'d'l\  of  which  d'  is  the  vertex  and  the  horizontal  line  through  c' 
is  the  axis. 

In  Fig.  210,  MN^  ^y^x^  ^re  two  positions  of  the  same  genera- 
trix, both  parallel  to  Y,  but  on  opposite  sides  of  the  surface ;  thus, 
on\  o'm^\  are  asymptotes  to  the  contour.  Draw  any  element  DF 
of  the  other  generation,  cutting  3f  N  at  B  and  M^N'^  at  S.  This 
element  pierces  the  meridian  plane  LL  at  P,  and  d's'  is  tan- 
gent at  p'  to  the  contour  (251.).  Now,  rp  =  ps,  consequently 
r'p'  =  p's' ;  that  is  to  say,  the  intercept  between  the  asymptotes  is 
bisected  at  the  point  of  contact :  which  is  a  property  peculiar  to  the 
hyperbola. 

Whence  it  appears  that  the  contour  lines  Tc'd'V ^  h'd'l\  in  Fig. 
^09,  are  the  opposite  branches  of  an  hyperbola  of  which  the  centre 
IS  o  and  the  major  axis  is  d'd' \  and  that  the  surface  may  be  gen- 
erated by  revolving  this  hyperbola  about  its  conjugate  axis.  This 
surface  being  unbroken,  is  distinguished  as  the  hyperboloid  of  revo- 
lution of  one  nappe  'y  the  same  hyperbola  by  revolving  about  th? 
major  axis  will  generate  an  hyperboloid  of  two  separated  nappes, — 
which  however  is  obviously  a  double-curved  surface. 

225a.  In  Fig  210a,  the  planes  TT^  LL^  parallel  to  Fand  to 
the  axis,  and  tangent  to  the  circle  of  the  gorge,  each  cut  from  the 
surface  two  rectiHnear  elements,  whose  vertical  projections  coincide 
and  are  asymptotes  to  the  meridian  hyperbola.     Draw  two  other 


172 


DESCRIPTIVE   GEOMETRY. 


planes  YY,  WW,  parallel  to  the  others  and  equidistant  from  the 
axis  but  nearer  to  it ;  the  sections  will  evidently  be  similar  curves, 
the  projections  of  those  on  the  right-hand  side  of  the  axis  coinciding 
in  ev'h'.  Any  rectilinear  element  of  the  surface,  as  IiS,  pierces 
these  planes  in  the  points  0,  0,  3f,  JV;  and  shice  oc  =  7mi,  it  fol- 


lows that  o'c'  =  m'nf ;  consequently  e^'N  is  also  an  hyperbola 
whose  asymptotes  are  the  same  as  those  of  the  meridian  outline. 

A  plane  parallel  to  these  but  farther  from  the  axis,  as  XX,  will 
cut  the  surface  in  an  hyperbola  whose  vertical  projection  g'f'k'  will 
have  the  same  asymptotes,  but  its  major  axis  will  be  vertical  instead 
of  horizontal. 

253.  In  Fig.  209,  the  first  position  of  the  generatrix,  MN\ 


DESCRIPTIVE    GEOMETRY.  173 

was  parallel  to  Y ;  but  in  applications  of  this  surface  as  an  auxiliary, 
the  given  generatrix  may  not  be  so  conveniently  situated.  To  illus- 
trate :  suppose,  in  Fig.  210,  tlie  vertical  axis  and  the  line  EG  to  be 
given,  from  which  d?.ta  the  surface  is  to  be  constructed.  Draw  ou 
perpendicular  to  eg ;  this  is  the  radius  of  the  gorge  circle,  the  alti- 
tude of  whose  plane  is  found  by  j)i*ojecting  i^  to  u'  on  eg' ;  this, 
done,  MN  is  drawn  parallel  to  T,  whence  the  asymptotes  are  deter- 
mined and  the  contour  constmcted  as  in  Fig.  209. 

If  we  regard  e'g'  in  Fig.  210  as  the  vertical  trace  of  a  plane 
perpendicular  to  T,  and  tangent  to  the  hyperboloid  at  K\  then  that 
plane  must  contain  the  companion  generatrix  passing  through  K^ 
whose  horizontal  projection  is  A'J;  also,  it  will  cut  the  meridian 
plane  LL  in  a  right  line  which  intersects  the  axis  of  the  surface  at 
Z,  and  it  is  ajDj^arent  that  KZ  is  the  true  tangent  at  K  to  the  hyper- 
bolic outline.  These  considerations  will  be  of  use  in  employing 
this  surface  as  an  aid  to  the  solution  of  the  following  problem. 

254.  Problem.  To  pass  through  a  given  right  line  a  plane 
tangent  to  any  given  surface  of  revolution. 

Analysis.  Revolve  the  given  line  about  the  axis  of  the  given 
surface,  thus  generating  a  warped  hyperboloid  of  revolution ;  the 
required  plane  will  be  tangent  to  this  hyperboloid  at  some  point  of 
the  given  line  (142).  It  will  also  be  perpendicular  to  the  meridian 
plane  through  the  point  of  contact,  on  either  surface ;  but  the  two 
surfaces  having  a  common  axis,  these  meridian  planes  will  coincide, 
and  will  cut  fi-om  the  common  tangent  plane  a  right  line,  tangent 
to  both  meridian  curves.  This  tangent  and  the  given  line  deter- 
mine the  required  plane. 

Construction.  In  Fig.  211,  let  CN  be  the  given  line,  through 
which  it  is  required  to  pass  a  plane,  tangent  to  the  surface  of  revo- 
tion  with  a  vertical  axis,  whose  contour  is  shown  in  the  vertical 
projection.  Find  the  radius  and  altitude  of  the  gorge  circle,  and 
construct  the  hyperboloid,  as  explained  in  (253).  Draw,  mechani- 
cally, s's'  tangent  to  the  given  contour  and  also  to  the  hyperbola 
li'g'V ;  this  is  the  vertical  trace  of  the  required  plane  when  revolved 
aboQt  the  axis  of  the  surface  until  it  is  perpendicular  to  V.  The 
plane  in  this  position  contains  an  element  C^D^  (the  revolved  po- 
sition of  the  given  line),  which  pierces  the  meridian  plane  LL  par- 


174 


DESCRIPTIVE   GEOMETRY. 


allel  to  V,  at  0^  the  revolved  position  of  tlie  point  of  contact  with 
the  hjperboloid.     The  revolved  position  of  the  point  of  contact 

X' 


FiG.  211 
with  the  given  surface  is  vertically  projected  at  j^/ ,  found  as  in 
(147),  and  horizontally  at  j^?,  on  LL, 

255.  In  the  counter-revohition,  O,  returns  in  a  horizontal 
circle  to  O  on  CD\  o  is  at  once  located  by  setting  off  do  =  d^o^ ; 
then,  drawing  the  radius  oe^  the  horizontal  trace  of  tlie  required 
plane  is  tcT  perpendicular  to  oe^  and  the  vertical  trace  is  Tt\ 
passing  through  n  ^  tlie  vertical  trace  of  the  given  line.  The  point 
of  contact  P  may  be  located  in  a  similar  manner  by  setting  off  on 
oe  the  distance  ep  —  ejp^ ;  the  vertical  projection  jp'  lies  upon  a  hori- 
zontal line  through  j^i'. 

Note.  The  common  tangent  line  PO^  mentioned  in  the  argu- 
ment, is  not  here  made  use  of,  since  the  directions  of  the  traces 
•can  be  determined  without  it,  and  in  this  instance  more  conven- 
iently ;  but  in  other  cases  the  use  of  that  line  might  be  essential. 

256.  Limiting  Forms  of  tliis  Surface.  If  in  Fig.  200  the  radius 
0C  be  gradually  diminished,  other  things  remaining  unchanged,  the 


DESCRIPTIVE    GEOMETRY.  175 

gorge  circle  will  become  less  and  less,  tlie  surface  assuming  a  closer 
resemblance  to  a  cone,  which  is  the  limiting  form  reached  when 
the  connnon  perpendicular  becomes  zero  and  the  generatrix  inter- 
sects tlie  axis. 

If,  oc  i-emaining  unchanged,  we  suppose  MN  to  make  a  greater 
angle  with  the  horizontal  plane,  the  curvature  of  the  surface  will 
diminish,  and  it  will  more  and  more  resemble  a  cylinder,  which  is 
the  limiting  form  reached  when  the  generatrix  is  parallel  to  the 
axis  :  the  diameters  of  tlie  gorge  and  the  base  circles  then  becoming 
equal. 

If  on  the  other  hand  the  generatrix  becomes  horizontal,  it  must 
always  lie  in  a  plane  perpendicular  to  the  axis,  which  therefore  is  a 
third  limiting  form  of  this  hyperboloid.  The  plane  generated  in 
this  manner,  however,  cannot  abstractly  be  considertjd  to  exist 
within  the  circumference  described  by  the  point  C\  wliicli  is  in 
fact  a  true  edge  of  regression. 

THE    ELLIITICAL    HYPERBOLOID. 

257.  Let  the  smaller  circle  E^  in  Fig.  212,  be  the  horizontal 
projection  of  the  gorge,  the  larger  one,  F^  that  of  the  upper  and 
lower  bases  equidistant  from  it,  and  m,n  that  of  the  generatrix, 
of  an  hyperboloid  of  revolution ;  whose  vertical  projection  is  con- 
structed as  above  explained.  If  the  ordinates  of  both  these  circles 
which  are  perpendicular  to  LL  be  reduced  in  the  same  proportion, 
the  circles  will  be  transformed  into  two  ellipses,  /  and  J\  which 
are  similar  to  each  other,  since  the  ratio  of  the  major  to  the  minor 
axis  is  the  same  in  both.  Now,  if  these  ellipses  be  substituted  for 
the  original  circular  gorge  and  bases,  a  right  line  moving  in  contact 
witli  them  all  will  generate  a  new  surface,  called  an  elliptical  hyper= 
boloid. 

If  we  constnict  as  in  Fig.  209  the  cone  directer  of  the  origina.. 
liyperboloid  of  revolution,  and  treat  its  base  in  like  manner,  that 
<3one  will  be  transformed  into  the  elliptical  cone  directer  of  the  new 
surface.  The  sections  of  this  cone  and  of  the  hyperboloid  itself  by 
parallel  planed  will  be  either  ellipses,  parabolas,  or  hyperbolas, 
according  to  the  positions  of  the  cutting  planes. 


176 


DESCKIPTITE    GEOMETRY. 


258,  The  chord  Tee  of  the  elhpse  «/,  tangent  to  the  ellipse  /  at 
the  extremity  of  the  minor  axis,  is  the  horizontal  projection  of  an 


Fig.  212 


element  of  this  surface :  this  element  is  by  construction  parallel 
and  equal  to  the  element  MN  of  the  hyperboloid  of  revolution ; 
and  will  have  the  same  vertical  projection,  since  mh^  ne^  are  per- 
pendicular to  T. 

Let  qf  be  the  horizontal  projection  of  any  other  element  of  the 
hyperboloid  of  revolution ;  it  is  tangent  at  o  to  the  circle  E^  and 
cuts  LL  in  p.  From  g  and  f  draw  ordinates  per23endicular  to  LL^ 
cutting  the  ellipse  Jin  c  and  d ;  and  from  o  draw  another,  cutting 
the  ellipse  I'm  r\  then  since  by  construction  gc^  fd,  or^  are  equal 
fractions  of  these  ordinates,  the  chord  cd  will  pass  through  7'  and^, 
and  by  a  known  property  of  the  curve  it  will  be  tangeir':  at  r  to  the 
smaller  ellipse. 

Consequently  cd  is  the  liorizontal  projection  of  an  element  o£ 


DESCRIPTIVE   GEOMETRY.  177 

tlie  elliptical  liyperboloid,  just  as  ^/is  of  an  element  of  tlie  circular 
one ;  and  since  gc^  fd^  are  perpendicular  to  V,  these  two  elements 
will  have  the  same  vertical  projection  c'd' :  which  is  also  the  ver- 
'tical  projection  of  another  element,  wdiose  horizontal  projection  i& 
drawn  through  ^,  tangent  to  the  gorge  ellipse  on  the  opposite  side. 

From  the  preceding  it  follows  that  the  elliptical  hyperboloid  is 
doubly  ruled,  and  that  wdien  placed,  as  here  shown,  witli  the  major 
axes  of  its  bases  parallel  to  V,  its  vertical  projection  will  be  identi- 
cal with  that  of  the  circular  one  from  which  it  is  derived  in  the 
manner  above  set  forth. 

Evidently,  the  ordinates  of  the  circles  might  have  been  in- 
creased, instead  of  being  diminished,  in   any  desired  proportion, 
without  affecting  the  argument :  in  which  case  the  minor  instead  of 
the  major  axis  would  have  been  parallel  to  T. 

In  the  profile,  it  is  evident  that  A/,  ^^,  will  be  equal  to  the 
minor  axis  of  the  larger  ellipse,  and  ab  to  that  of  tiie  smaller  one. 
The  contour  in  this  view  may  logically  be  determined  thu§ :  Draw 
xyz  the  contour  of  the  original  circular  hyperboloid ;  then  ordinate^ 
from  this  contour  perpendicular  to  the  axis  are  to  be  reduced  in 
the  same  proportion  that  was  adopted  in  constructing  the  ellipses  / 
and  J.  Treating  in  the  same  way  the  asymptotes  st^  su,  to  the 
curve  Xi/2,  we  shall  have  two  right  lines,  ww,  vv ;  which  are  the 
two  elements  represented  by  wv  in  the  horizontal  projection.  From 
this  construction  it  is  seen  that  sw,  sv,  are  apymptotes  to  the  con- 
tour/^ta^;  which  can  be  shown  to  be  an  hyperbola  by  reasoning 
similar  to  that  of  (252). 

260.  To  assume  a  point  upon  this  surface.  If  the  horizontal 
projection  be  assumed,  the  horizontal  projection  of  an  element 
through  the  point  w^ill  be  a  tangent  to  the  gorge  ellipse.;  and  the 
vertical  projection  of  the  point  must  lie  upon  the  vertical  projec- 
tion of  this  element :  it  can  therefore  be  determined  directly. 

But  if  the  vertical  projection  be  assumed,  as  at  S^ ;  it  is  then 
necessary  to  draw  the  section  of  the  surface  by  a  plane  JTX  passing 
through  the  point  and  perpendicular  to  the  axis.  It  follows  from 
what  precedes  that  this  section  will  be  an  ellipse  of  which  the  major 
axis  is  Gllm  the  vertical  projection,  and  the  minor  axis  is  DQ  in 
the  profile.     This  ellipse  will  be  seen  in  its  true  form  in  the  hori- 


178  DESCKIITIVE    GLUMETllY. 

zontal  projection,   and  tlie  horizontal  projection  of  the  assumed 
point  must  Re  upon  tlie  curve. 

To  draw  a  plane  tangent  to  this  surface  at  a  given  point.  The 
surface  being  doubly  ruled,  the  tangent  plane  is  determined  by 
drawing  through  the  point  an  element  of  eacli  generation. 

261.  If,  in  Fig.  212,  the  ordinates  perpendicular  to  LL  in  the 
horizontal  projection  liad  been  increased  instead  of  diminished  in 
any  given  ratio,  the  circles  would  have  been  transformed  into 
elHpses  with  their  minor  axes  parallel  to  V.  But,  the  preceding 
argument  still  holding  true,  the  vertical  projection  of  the  resulting 
elliptical  hyperboloid  would  liave  been  identical  with  that  of  the 
original  circular  one  :  accordingly,  the  profile  in  the  figure  may  be 
regarded  as  the  vertical  projection  of  an  hyperboloid  of  revolution, 
in  which  the  diameters  of  the  gorge  and  the  bases  are  respectively 
equal  to  the  minor  axes  of  the  ellipses  1  and  «/;  as  indicated  in  the 
diagram  B,  drawn  below  the  profile. 

It  follows,  then,  not  only  that  from  a  given  circular  hyperboloid 
an  infinite  number  of  elliptical  ones  may  be  derived ;  but  that  the 
former  is  merely  a  special  case,  in  which  the  axes  of  the  three  ellip- 
tical directrices  become  equal. 

262.  Limiting  Forms  of  tlie  Elliptical  Hyperboloid.  By  rea- 
soning analogous  to  that  of  (256),  it  will  be  apparent  that  when  the 
generatrix  intersects  the  axis,  the  surface  will  become  an  elliptical 
cone ;  when  it  is  parallel  to  the  axis,  the  surface  will  become  an 
elliptical  cylinder ;  and  if  it  is  tangent  to  the  gorge  ellipse,  the  sur- 
face will  become  a  plane  with  ati  elliptical  perforation. 

263.  Tlie  Hyperbolic  Paraboloid  a  Special  Case.  A  limiting  form 
may  be  reached,  however,  in  a  different  manner.  In  Fig.  213, 
let  c  be  the  axis  and  rarn  the  generatrix  of  an  hyperboloid  of  revo- 
lution ;  the  latter  retaining  its  present  position,  suppose  the  axis  to 
recede,  as  indicated  by  the  successive  positions  <?, ,  c^ :  it  is  clear  that 
the  difference  between  the  radii  cr  and  cm  will  become  less  and 
less,  until  when  the  axis  is  infinitely  remote,  those  radii  will  be 
equal,  the  two  circumferences  will  become  one  right  line,  and  the 
surface  will  become  a  plane ;  which  therefore  is  a  Ihniting  form  in 
this  case. 

In  Fig.-  214.,  the  gorge  and  base  ellipses  having  been  derived 


DESCRirTlVE   GECMETRT. 


179 


from  tlie  circles  shown,  as  in  Fig.  212,  let  ^  and  e  be  their  corre- 
sponding foci :  then  from  the  mode  of  derivation  it  follows  that  the 
semi-parameters  os^  er\  have  the  same  ratio  as  the  radii  ca^  cb. 
Kow,  the  points  <^,  5,  and^*  remaining  fixed,  suppose  again  the  axis 

The  curves,  as  has  been  seen,  will  always 


to  recede  to 


'15    ^a)   ^^^' 


be  similar  ellipses;  and  their  semi-parameters  approach  equality  as 
the  axis  recedes,  becoming  equal  when  its  distance  is  infinite :  but 
at  this  limit,  the  conjugate  focus  o'  being  also  infinitely  remote,  the 
gorge  ellipse  becomes  a  parabola;  and  from  the  preceding  argu- 
ment it  follows  that  tlie  outer  ellipse  will  also  become  a  parabola 
liamng  tlie  same  parameter.  In  these  circumstances,  the  elliptical 
hyperboloid  becomes  a  hyperbolic  paraboloid;  which  in  view  of 
this  mode  of  derivation  might  more  consistently  be  called  a  para- 
bolic hyperboloid. 

264.  It  will  now^  be  clear  that  the  gorge  plane  PP^  and  the 
plane  /.//,  in  Fig.  212,  correspond  in  position  to  the  principal  di- 
ametric planes  in  Figs.  196-199;  and  supposing  the  axis  of 'the 
surface  to  recede  to  the  left,  the  gorge  ellipse  /will  at  the  limit 
become  the  gorge  parabola  of  Fig.  198,  and  the  hyperboloidal  con- 

*  As  c  recedes,  os  will  increase,  while  he  and  er  will  diminish  ;  until,  when  ac 
becomes  infinite,  we  shall  have  he  —  ao,  and  er  =  os  =  2ao, 


180 


DESCRIPTIVE    GEOMETRY. 


tonr  HOp'  will  become  the  other  gorge  parabola,  shown  in  Fig. 
197;  O^  the  connnon  vertex  of  the  ellipse  and  the  hyperbola  iu 
Fig.  212,  will  remain  the  connnon  vertex  of  the  two  gorge  parab- 
olas :  it  will  also  be  the  vertex  of  the  resulting  hyperbolic  parabo- 
loid,— whose  axis,  however,  has  no  relation  to  the  axis  of  the  original 
surface,  since  it  is  merely  the  common  axis  of  the  two  gorge  lines, 
being  the  intersection  of  tlie  planes  PP^  LL. 

265.  The  ellipse  «/,  in  Fig.  212,  represents  the  intersection  of 
the  elliptical  hyperboloid  by  either  of  two  planes,  parallel  to  PP 
and  equidistant  from  it.  And  we  observe  that  any  chord  cd  of 
this  ellipse  which  is  tangent  to  the  smaller  one  /,  is  bisected  at  r^ 
the  point  of  contact ;  which  from  the  mode  of  derivation  must 
hold  true  whatever  the  relative  magnitudes  of  the  corresponding 
axes, — and  these  again  are  independent  of  the  distances  from  the 
gorge  plane. 

At  the  limit  both  ellipses  become  parabolas ;  whence  the  fol- 
lowing   problem    may    arise.     Given,   in  Fig.   215,  the   parabola 


Fig.  215 


HOK^  of  which  O  is  the  vertex  and  OR  the  axis :  Required,  to 
construct  another  parabola  with  the  same  axis,  and  a  given  vertex 
A.^  any  chord  of  which,  tangent  to  the  given  one,  shall  be  bisected 
at  the  point  of  contact. 

Any  parabola  having  the  same  axis  will  satisfy  the  condition 
for  the  chord  tangent  at  0'^  it  remains  then  to  find  one  which  will 
satisfy  it  in  regard  to  some  other  chord.  Set  off  OB  =  OA^  erect 
the  ordinate  BC\  draw  J.  6^  tangent  to  TlOIy^  produce  it,  and  make 
CD  =  CA ;   then  Z>  is  a  point  in  the  required  curve  PAL     Draw 


DESCRIPTIVE    GEOMETRY. 


181 


tlie  ordinates  DE^  OG^  then  by  construction  AE=  2AB  =  4:A0, 
and  since  the  curve  is  a  parabola,  DE=  2 GO]  but  by  construc- 
tion DE=  2BC,  consequently  GO  —  BC.  That  is  to  say,  the  two 
parabolas  are  precisely  alike;  which  agrees  with  the  conclusion 
otherwise  reached  in  (263). 

2.66,  In  Fig.  216,  tlie  two  parabolas  Tiok^  dai,  in  the  horizontal 
projection,  are  the  same  as  those  of  the  preceding  diagram;  the 


"K^ 

^^ 

m 

^^v 

^.S' 

> 

^ 

V 

^ 

\ 

r. 

\ 

"yC 

% 

z 

i 

^ 

0 

yn 

ja 

vertical  projection  of  the  former  is  in  the  line  PP,  while  s'd',  a'i\ 
parallel  to  PP  and  equally  distant  from  it,  are  the  vertical  projec- 
tions of  the  latter.  Kow,  whatever  the  actual  magnitudes  of  x's' 
and  x'a\  it  is  to  be  observed  that  s'i'^  a'd\  will  intersect  at  y'  on 
PP^  giving  y'o'  =  o'x\  and  will  therefore  be  tangent  to  a  parabola 
of  which  o'  is  the  vertex ;  and  in  like  manner  it  may  be  sliown  that 
m^n'^  w'z\  and  in  short  the  vertical  projections  of  all  tangents  to 
liok^  will  be  tangent  to  the  same  parabola  a'o's' ;  which  lies  in  the 
vertical  plane  LL :  the  surface  thus  determined  is,  then,  2.  hyper- 
bolic paraboloid. 


183  DESCRIPTIVE    GEOMETRY. 

Bj  similar  reasoning  it  may  be  sliown  tliat  sections  hy  planes 
parallel  to  ZL,  will  be  parabolas  precisely  like  a'(/s' ;  that  is  to  say, 
all  sections  of  tins  surface  by  planes  parallel  to  either  of  the  princi- 
pal diametric  planes,  will  be  similar  and  equal  to  the  gorge  parab- 
ola which  lies  in  that  plane. 

267.  If  we  suppose  the  parameter  of  the  gorge  hok  to  be 
indefinitely  reduced,  other  things  remaining  unchanged,  that  pa- 
rabola will  ultimately  become  c,  right  line,  and  the  surface  Mill 
become  a  plane  coinciding  with  ZZ^  its  elements  being  then  tan- 
gents to  ao's.  Similarly,  if  the  -horizontal  gorge  remain  un- 
changed, while  the  parameter  of  a'o's'  is  reduced,  the  elements  will 
"ultimately  become  tangent  to  hok,  and  the  surface  a  plane  coincid- 
ing with  Z*P. 

268.  This  derivation  of  the  hyperbolic  jjaraboloid,  though 
showing  it  to  be  doubly  niled,  gives  thus  far  no  evidence  of  the 
existence  of  plane  directers.  But  if  it  be  established  tliat  the  ele- 
ments of  each  generation  divide  those  of  the  other  proportionally, 
it  must  follow  that  those  of  either  set  lie  in  a  series  of  parallel 
planes. 

If  this  relation  is  true  as  to  the  elements  in  space,  it  must  also 
be  true  of  their  projections,  in  which  all  j)arts  are  equally  fore- 
shortened ;  in  other  words  the  tangents  to  the  gorge  parabolas  in 
Fig.  216  must  divide  each  other  proportionally.  That  they  do  so, 
is  a  fact  made  use  of  in  the  familiar  construction  shown  in 
Fig.  217 ;  the  sides  J^C,  CD,  of  the  isosceles  triangle  ZJCIJ  being 
divided  into  tlie  same  number  of  equal  parts,  the  points  of  division 
are  numbered  in  opposite  directions,  and  the  corresponding  num- 
bers joined  by  right  lines :  the  envelope  of  these  lines  is  the  parab- 
ola EAD.  The  following  demonstration  of  this  depends  upon  two 
properties  of  the  parabola,  viz.,  that  the  abscissas  are  proportional 
to  the  squares  of  the  ordinates,  and  that  the  subtangent  is  bisected 
at  the  vertex. 

269.  In  Fig.  218,  let  A  be  the  middle  point  of  BC\  which 
bisects  the  vertical  angle  of  the  isosceles  triangle  EDC.  On  the 
equal  sides  set  off  DF  —  Oil  of  any  length  at  pleasure,  draw  ZII 
cutting  ^^'in  G,  and  on  JjO  set  oif  AM  —  AG.     Perpendicuki 


DESCRIPTIVE    GEOMETRY. 


183 


to  BCdmw  FO,  ML  cutting  FH'm  Z,  ^P  cutting  FH'm  iTand 
I) Cm  P,  and  ///cutting  BCin  iV^and  CD  in  / 

Kow  since  BCis  bisected  at  A^  CD  is  tangent  at  />  to  a  pa- 
rabola of  which  A  is  the  vertex ;  and  since  MG  is  bisected  at  A, 
FJIis  tangent  at  Z  to  a  parabola  of  which  A  is  the  vertex.  Also 
these  parabolas  have  the  common  axis  DC',  but  in  order  to  prove 


F      D 


Fig.  2L7 


Fio.  21S 


that  they  are  one    and  the  same,  it  is   necessary  to   show  that 
AM\  AB  ::  MD  :  BD\ 

But  AM^  AG,  AB  =  AC,  and   ML  :  BD  ::  AK \  AP-, 
consequently  the  above  proportion  may  he  written  AG  :  AG  :: 

AK'  :  AF^\   or,  in  fractional  form,  -j-^  =  ~TW^' 

In  order  to  demonstrate  this,  we  proceed  as  follows:  Draw 
througli  F 'd  parallel  to  DC,  cutting  BD  in  F' ,  and  ML  produced, 
in  F''-,  draw  also  NF',  NF",  Then  since  FF'  ^  CN,  NF'  ie 
parallel  to  CD\  and  since  FF"  =  NG,  NF"  is  parallel  to /7Z 
Also,  since  FH,  FL,  are  bisected  by  J./*,  we  have  ZTA^  =  iV/ 
=  KP,  =  F'D,  =  LF" ;  therefore  NF'  passes  through  K,  and 
NF"  passes  through  P. 


184  DESCIIIPTIVE    GEOMETRY. 

Then  from  similar  triangles  GAK,  NAP,  we  liave  ~,  =  ^~    I 

and       ''         ''  "       JVAK,OAP,"      "     ^=^\l^,\ 

AG  _  AK' 
•*•  AC-AP^'  ^•^•■^• 

269a.  These  hjperboloids,  viz.,  the  circular,  elliptic,  and  para- 
bolic, are  the  only  doubly-ruled  warped  surfaces,  and  also  the  only 
ones  having  three  rectilinear  directrices.  Each  has  two  gorge  lines, 
lying  in  planes  perpendicular  to  each  other,  and  having  a  common 
vertex ;  and  from  the  preceding  it  is  apparent  that  if  sections  be 
made  by  planes  parallel  to  and  equidistant  from  either  gorge,  their 
projections  on  the  plane  of  that  gorge  must  coincide,  be  similar  to 
the  gorge  curve  and  similarly  placed,  and  satisfy  the  condition  that 
any  chord  of  the  outer  curve,  tangent  to  the  inner,  shall  be  bisected 
at  the  point  of  contact.  It  has  been  shown  that  this  condition  can 
be  satisiied  by  the  conic  sections ;  and  it  is  not  known  that  it  can 
be  satisiied  by  any  other  curves. 

THE    HELICOID. 

270.  \Ye  shall  use  the  term  helix  in  the  restricted  sense  in  which 
it  is  commonly  employed ;  as  designating  the  path  of  a  point  which, 
w^hile  revolving  uniformly  around  an  axis,  also  moves  uniformly  in 
a  direction  parallel  thereto. 

This  curve,  then,  lies  upon  the  surface  of  a  cylinder  of  revolu- 
tion, cuts  all  its  rectilinear  elements  at  the  same  angle,  and  becomes 
a  right  line  when  tlie  cylinder  is  developed  into  a  plane. 

This  being  understood,  we  shall  use  the  term  Helicoid  to  desig- 
nate any  surface  generated  by  a  right  line  whose  motion  is  deter- 
mined by  helical  directrices  lying  upon  concentric  cylinders. 

The  right  line,  thus  controlled,  must  necessarily  have  a  motion 
of  revolution ;  and  it  may  either  intersect  the  axis,  like  the  genera- 
trix of  a  circular  cone,  or  remain  always  at  a  fixed  distance  from 
it,  like  that  of  an  hyperboloid  of  revolution. 

271.  Helicoidsof  Uniform  Pitch  and  of  Tarying  Pitch.  If  all 
the  helical  directrices  have  the  same  pitch,-  every  point  of  the  gen- 
eratrix will  travel,  in  a  direction  parallel  to  the  axis,  at  the  same 


DESCRIPTIVE   GEOMETRY. 


185 


rate ;  and  that  line  will  either  lie  in,  or  make  a  constant  angle 
with,  a  plane  perpendicular  to  the  axis :  the  surface  is  then  said  to 
be  of  uniform  pitch. 

But  the  directrices  may  be  of  different  pitches ;  in  which  case, 
the  rate  of  arial  advance  being  different  for  the  various  points  of 
the  generatrix,  that  line  will  continually  change  its  inclination  to 
the  transverse  plane ;  and  the  surface  is  then  called  a  helicoid  of 
varying  pitch. 

272.  Right  and  Ohlique  Helicoids.  The  hehcoids  of  uniform 
pitch  may  be  subdivided  into  two  classes.  In  Figs.  219  and  220, 
let  the  generatrix  DE  vqyo\yq  uniformly  around  the  vertical  axis, 


Fig.  ^9 


Fig.  220 


while  at  the  same  time  all  its  points  move  uniformly  in  a  direction 
parallel  to  the  axis;  then  the  point  C'will  describe  tha  helix  MCN 
lying  on  the  inner  cylinder,  and  the  points  D  and  E  will  describe 
helices  of  the  same  pitch  lying  on  the  outer  cylinder ;  of  which 
r's't'^  xy'u\  are  the  vertical  projections.  In  both  cases,  the  gen- 
eratrix remaining  at  the  fixed  distance  oc  from  the  axis,  will  always 
lie  in  a  plane  tangent  to  the  inner  cylinder,  touch  that  cylinder  in 
a  single  point,  and  cut  the  vertical  element  through  that  point  at  a 
constant  angle. 

In  Fig.  219,  the  generatrix  is  jperjpendicular  to  that  element^ 


186  DESCRIPTIVE    GEOMETRY. 

and  therefore  parallel  to  the  transverse  plane ;  that  plane,  then,  h 
the  plane  directer  of  the  surface,  which  is  called  a  right  helicoid. 
The  radius  oc  being  arbitrary,  may  be  reduced  to  zero ;  in  which 
case  the  generatrix  intersects  the  axis,  and  the  helicoid  becomes  a 
special  form  of  the  right  conoid,  the  axis  being  the  line  of  striction : 
this  is  a  most  familiar  modification  in  practice,  since  it  is  the  sur- 
face of  the  common  square-threaded  screw. 

In  Fig.  220,  the  generatrix  is  inclined  to  the  vertical  element 
of  the  inner  cylinder  through  the  point  of  contact.  The  surface  is 
then  called  an  oblique  helicoid,  and  has  a  cone  directer  instead  of  a 
plane  directer :  if  ^c  be  reduced  to  zero  in  this  case,  the  surface  is 
that  of  the  ordinary  triangular  or  Y-threaded  screw. 

273.  Representation  of  the  Helicoid.  Since  each  of  the  points 
2>,  6^,  E\  in  the  two  preceding  figures,  describes  a  helix  of  the 
same  pitch,  it  is  easy  to  draw  as  many  elements  as  may  be  desired. 
But  the  mere  projection  of  a  number  of  elements  of  indefinite 
length  does  not  ordinarily  convey  an  adequate  idea  of  the  form  of 
any  surface.  This  is  better  done  by  representing  a  limited  portion, 
as  in  Fig.  221 :  the  conditions  here  are  similar  to  those  in  Fig.  220, 
and  the  surface  is  at  once  recognized  as  resembling  that  of  the 
groove  in  an  auger  or  a  twist  drill.  This  figure  illustrates  the 
principle  already  mentioned,  that  the  visible  contour  is  the  envelope 
of  all  lines  of  a  surface ;  it  is,  accordingly,  tangent  to  the  helical 
paths  of  the  points  Jf,  AT,  etc. ,  which  are  used  for  this  determina- 
tion in  preference  to  the  rectilinear  elements :  the  latter  would  of 
course  have  served  the  same  purpose,  but  practically  would  have 
been  more  confusing  and  very  little  if  any  easier  to  construct. 

274.  To  assume  a  point  on  the  surface.  Supposing  the  axis  to 
be  vertical,  as  shown ;  then  if  the  horizontal  projection  be  assumed,, 
an  element  can  at  once  be  drawn  through  the  point.  If  the  verti- 
cal projection  be  assumed,  the  point  must  lie  on  a  perpendicular  to 
V  through  the  point ;  any  plane  through  this  line  will  cut  the  ele- 
ments in  points  determining  a  curve,  whose  liorizontal  projection 
"will  cut  that  of  the  line,  in  the  required  horizontal  projection  of 
the  point. 

275.  Peculiarities  of  Meridian  Sections.  If  in  Fig.  221  the  sur- 
face be  cut  by  a  plane  through  the  axis,  perpendicular  to  T,  the 


DESCRIPTIVE   GEOMETRY. 


187 


section  will  be  the  curve  gfji^ ,  shown  in  the  profile.  This  curve 
is  easily  constructed  by  drawing  elements  of  the  surface,  and  finding 
the  poiuts  in  which  they  pierce  the  cutting  plane :  for  example, 
the  element  XY  pierces  that  plane  at  P ;  the  altitude  of  jy,  is  the 
samt  as  that  of  p' ^  and  its  distance  jp^o^  from  the  axis  is  equal  to  po. 


This  curve  is  convex  toward  the  axis,  and  tangent  at  its  vertex  <?, 
to  the  vertical  element  of  the  inner  cylinder.  This  helicoid,  then, 
surrounds  a  cylindrical  well,  within  which  it  cannot  extend ;  and 
since  all  the  meridian  sections  are  obviously  alike,  the  surface  is 
tangent  to  the  cylinder  all  along  the  helix  KCL,  which  is  a  true 


In  Fig.  222,  the  generatrix  is  perpendicular  to  the  element  of 
the  inner  cylinder  through  the  point  of  contact,  as  in  Fig.  219-' 
the  diameters  of  the  two  cylinders,  as  well  as  the  helical  pitch,  are 


188 


DESCRIPTIVE    GEOMETRY. 


the  same  as  in  Fig.  221.     This  change  in  the  position  of  the  gen- 
*^ratrix  lias  caused  the  helices  described  by  D  and  E  to  approach 


each  other,  so  that  the  breadth  g'h!  of  the  groove  is  less  than  be- 
fore. 

Still  the  form  of  the  meridian  section  is  not  greatly  changed ; 
and  it  is  evident  upon  consideration  that  if  the  generatrix  be 
lengthened,  thus  enlarging  the  outer  cylinder,  the  curve  gcji^  will 
continue  to  expand  as  it  recedes  from  the  vertex  c, ,  though  less 
and  less  rapidly;  the  curve  having  two  horizontal  asymptotes 
through  Z, ,  ^j . 

276.  In  Fig.  223,  T>CE\%  again  inclined,  but  in  the  opposite 


DESCRIPTIVE    GEOMETRY. 


189 


direction ;   and  to  such  an  extent  that  D  now  lies  upon  the  lower 
and  E  upon  the  higher  of  the  two  helices  on  the  outer  cylinder. 


The  result  is  that  the  meridian  section  is  a  looped  curve,  crossing 
itself  at  a  point  u^  between  the  two  cylinders.     Thus  this  helicoid 


190  DESCRIPTIVE    GEOMETRY. 

encloses  not  only  the  cylindrical  wall,  to  which  it  is  tangent  along 


Fig.  224  a 

the  gorge  helix  KCL^  but  also  a  serpentine  void  whose  transverse 
section  is  of  the  form  of  the  loop  xi^w^c^z^ .  ' 

In  these  circumstances,  a  groove  of  the  form  li{a^g^  can  be  cut 


DESCRIPTIVE   GEOMETRY.  191 

in  the  outer  cylinder,  bounded  by  helicoidal  surfaces  of  which  DE 
is  tlic  generatrix :  but  the  actual  formation  of  the  surface  below 
the  point  ?/,  is  impracticable. 

277.  In  Fig.  223,  the  angle  d'c'Ii'^  between  i>^ and  the  verti- 
cal element  of  the  inner  cylinder  through  C^  is  nevertheless  greater 
than  that  between  this  element  and  the  tangent  to  the  helix  KCL. 
]S^o\v  in  Fig.  224,  these  angles  are  equal ;  in  other  words,  the  gen- 
eratri  x  DE  is  tangent  at  G  to  the  helix :  the  surface,  although  an 
oblique  helicoid,  is  now  developable,  being  in  fact  the  helical  con- 
volute previously  discussed. 

In  the  meridian  section,  the  loop  has  now  disappeared,  u^  hav- 
ing retreated  to  c, ,  at  which  point  the  curve  g^cji^  forms  a  cusjpy 
the  tangent  to  this  curve  at  this  point,  moreover,  is  horizontal  in- 
stead of  vertical,  and  the  helix  KCL  is  no  longer  a  gorge  line  but  an 
edge  of  regression ;  along  which  the  helicoid  intersects  the  cylinder, 
to  which  it  is  not  tangent,  as  has  been  erroneously  stated. 

Finally,  in  Fig.  225  the  angle  dldh'  is  less  than  the  angle  be- 
tween the  tangent  to  KCL  and  the  vertical  element  through  C, 
We  now  have  again  as  the  meridian  section  a  curve  whose  tangent 
at  its  vertex  <?,  is  an  element  of  the  inner  cylinder ;  and  as  before 
the  helix  Is^CL  is  a  gorge  line,  along  which  the  helicoid  is  tangent 
to  that  cylinder. 

278.  Peculiarities  of  TransTerse  Sections.  The  transverse  sec- 
tion of  any  helicoid  may  be  determined  in  the  ordinary  manner,  by 
finding  the  points  in  which  any  number  of  elements  pierce  the  cut- 
ting plane.  But  though  perfectly  correct  in  theory,  this  method 
is  in  this  case,  as  in  many  others,  practically  more  laborious  and  far 
less  reliable  than  that  of  constructing  the  curve  by  the  aid  of  other 
known  properties. 

Tims,  let  it  be  required  to  draw  the  transverse  section  by  a 
plane  through  (7,  in  Fig.  224.  The  axis  being  vertical,  this  plane 
is  horizontal,  and  tlie  required  section  TcT'  is  no  other  than  the 
horizontal  trace  of  the  surface ;  and  this  (128)  is  the  involute  of  the 
circular  base  of  the  cylinder  upon  which  the  helix  KCL  is  traced. 
Drawing  tangents  at  5,  Z,  «,  etc. ,  set  off  upon  them  hp  —  arc  cb^ 
Im  =  arc  cl,  and  so  on ;  the  curve  passing  through  the  points  thus 
located  is  the  required  section :  the  distance  rs^  set  off  upon  the 


192 


DESCRIPTIVE    GEOMETRY. 


most  remote  tangent,  is  in  this  case  equal,  of  conrse,  to  the  semi- 
circumference  of  the  base. 

N.  B.  Tliis  curve  forms  a  cusp  at  C^  as  did  the  meridian  section ; 
and  so  will  the  section  of  this  particular  surface  by  any  other  plane 


Fig,  225 


through  the  same  point,  with  the  exception  of  those  planes  which 
pass  through  the  generatrix  DE. 

279.  The  transverse  section  of  the  surface  shown  in  Fig.  223 
will  not  be  a  true  involute,  because  the  generatrix  is  not  tangent  to 
the  helix  KCL ;  but  it  may  be  constructed  in  a  manner  analogous 
to  that  above  described.  In  Fig.  223a,  let  c'l'r'  be  a  half  turn  of 
the  helix ;  draw  through  c'  a  horizontal  line  c  8  ^  and  through  r' 
the  vertical  projection  of  an  element  of  the  helicoid;  then  o'^' is 
evidently  the  distance  to   be  set  off  as  rs  on  the  most  remote  tan- 


DESCRIPTIVE    GEOMETRY.  193 

gent.  The  motion  of  tlie  generatrix  consists  of  a  uniform  rotation, 
combined  with  a  unifoi-m  axial  advance;  consequently,  dividing 
c's'  in  Fig.  223<2  into  equal  parts  by  the  points  1,  2,  3,  and  the 
8emi- circumference  clr  into  the  same  number  of  equal  parts,  we 
draw  tangents  at  the  points  of  subdivision  5,  Z,  a,  and  set  off  upon 
them  the  distances  hi  =  c'l,  12  —  c'2,  a^  —  c'Z  :  the  curve  passing 
tlirough  the  points  1,  2,  3,  s,  is  the  section  required.  As  many 
intermediate  points  as  may  be  deemed  necessary  can  be  located  in 
like  manner;  and  the  resulting  curve  in  this  case  will  be  looped, 
crossing  itself  at  u,  the  distance  uo  being  equal  to  u^o^  in  the  me- 
ridian section. 

280.  The  formation  of  this  loop  in  Fig.  223  is  obviouslj  due 
to  the  fact  that  the  distances  hi,  Z2,  etc.,  set  off  on  the  tangents, 
are  greater  than  the  arcs  ch,  d,  etc.  In  Fig.  225,  on  the  other 
hand,  these  distances,  determined  in  the  same  manner,  are  less 
than  the  corresponding  arcs.  The  result  is  that  the  curve  TcT  has 
a  wave  instead  of  a  loop :  and  ^t  is  observed  that  the  meridian 
section  also  possesses  this  distinguishing  feature. 

In  thus  passing  from  the  loop  of  Fig.  223  to  the  cusp  of  Fig. 
224  and  the  wave  of  Fig.  225,  the  angle  d'c'h^  has  been  pro- 
gressively diminished.  If  on  the  other  hand  that  argle  be  in- 
creased, the  loop  will  become  larger,  the  curvature  at  c  growing 
less  and  less,  until  when  the  angle  reaches  the  limit  of  90°,  as  in 
Fig.  222,  there  will  be  no  curvature  at  all,  and  the  transverse  sec- 
tion will  be  simply  the  generatrix  D^  itself.  If  this  limit  be 
passed  and  the  angle  d'c'h'  made  obtuse,  as  in  Fig.  221,  the  section 
TcT,  still  tangent  at  0  to  that  of  the  inner  cylinder,  will  curve  in 
the  opposite  direction. 

281.  Intersections  of  the  Oblique  Helicoid  with  Itself.  The 
plane  PP,  in  Fig.  224«^,  is  tangent  to  the  inner  cylinder,  upon 
which  lies  the  helical  directrix  of  the  developable  helicoid  repre- 
sented by  this  figure.  This  plane  cuts  the  surface  in  the  genera 
ivix  UPS,  and  also  in  the  curve  WPT,  as  shown  in  the  side  view. 
After  one  revolution,  ^-^S'  will  be  found  at  rs,  after  another  at  r's\ 
and  so  on ;  and  in  these  new  positions  will  cut  the  curve  PT  in 
the  points  5,  s\  s'\  etc.  The  point  s  describes  a  helix  xx,  lying  on 
a  cylinder  of  which  aa  is  an  element ;  the  point  6''  traces  the  helix 


194  DESCRIPTIVE    GEOMETRY. 

yy^  on  a  cylinder  of  wliicli  JA)  is  an  element ;  tlie  jDoint  s"  in  like 
manner  traces  a  helix,  not  shown,  on  a  cylinder  of  which  cc  is  an 
element — and  so  on  indefinitely ;  and  these  helices,  all  having  the 
same  pitch  as  the  original  directrix  on  the  inner  cylinder,  are  evi- 
dently the  lines  of  intersection  between  the  successive  convolutions 
of  the  surface. 

And  it  is  obvious  that  as  shown  in  the  figure  a  screw  may  be 
cut,  the  crest  of  the  thread  being  the  helix  xx^  and  its  root  being 
the  directrix :  or  a  larger  one  of  the  same  pitch,  the  thread  having 
yy  for  its  crest  and  xx  for  its  root ;  these  screws  will  be  Hivgle- 
threaded^  and  their  surfaces  are  portions  of  the  same  helicoid. 

Fig.  224,  on  the  other  hand,  represents  a  doidjle-threaded 
screw;  as  seen  in  the  side  view,  two  helices,  k'c'l\  f'q't\  are 
traced  on  the  inner  cylinder,  to  wdiich  the  two  generatrices,  d'c'e\ 
f''it\  are  respectively  tangent :  these  lines  thus  generate  two  dis- 
tinct helicoids,  which  intersect  each  other  in  the  helices  d'h'u\ 
v'g'e\  forming  the  crests  of  the  threads. 

282.  That  the  surface  of  a  warped  helicoid  will  also  intersect 
itself  in  a  series  of  helices  is  shown  in  Fig.  225 ;  a  plane  through 
DE^  parallel  to  the  axis,  cuts  the  surface  in  the  curve  [fc'FJ^ 
which,  however,  is  not  tangent  to  the  generatrix,  but  crosses  it  at 
the  point  C.  After  one  revolution,  this  generatrix  will  have  the 
position  rs^  cutting  the  curve  at  s^  which  point  will  describe  a  helix 
corresponding  to  xx  in  the  preceding  figure ;  and  successive  ones 
may  be  determined  in  the  manner  above  explained. 

In  the  case  of  the  right  helicoid,  however,  the  plane  tangent  to 
the  inner  cylinder  cuts  the  surface  in  the  generatrix  only ;  and 
since  this  is  parallel  to  its  position  after  any  given  number  of  com- 
plete revolutions,  this  surface  does  not  intersect  itself  like  the 
others,  no  matter  how  far  extended. 

283.  Special  Case  of  the  Right  Helicoid.  In  Fig.  226,  the  gen 
eratrix  intersects  the  axis  at  right  angles,  and  the  rotation  being 
in  the  direction  indicated  by  the  arrow  in  the  end  view,  the  point 
d  describes  the  helix  def^  while  g  describes  the  helix  ghh ;  any  other 
points  of  the  generatrix,  as  x,  s,  will  describe  lielices  of  different 
obliquities,  since  they  lie  upon  cylinders  af  different  diameters,  but 
because  they  all  have  the  same  pitch,  these  helices  will  in  the  side 


DESCRIPTIVE    GEOMETRY. 


195 


view  intersect  in  the  points  s,  s\  etc. ,  lying  on  the  projection  of 
the  axis.     This  figure  represents  an  ideal  single- threaded  screw, 


Fig.  226 


Fig.  228 


which  may  be  conceived  as  being  formed  by  uniformly  twisting  a 
thin  rectangular  strip  of  flexible  metal  about  its  longitudinal  centre 
line.     Suppose  two  such  strips  to  be  fastened  together  at  right 


196  DESCRIPTIVE    GEOMETRY. 

angles,  presenting  when  viewed  endwise  the  form  of  the  cross,  dg^ 
mn,  Fig.  227 ;  let  each  be  iiniformlj  twisted  in  the  same  direction 
and  at  the  same  rate  as  in  the  preceding  figure :  then  711  will 
describe  the  helix  mlq^  while  n  describes  the  helix  nj[)r^  and  the 
result  will  be  the  formation  of  the  surface  of  an  ideal  double- 
threaded  screw.  The  practical  application  is  shown  in  Fig.  228, 
in  the  construction  of  a  single-threaded  screw,  cut  upon  a  cylinder 
whose  radius  is  od^  the  depth  of  the  groove  being  limited  by  the 
central  ' '  core, ' '  who^e  radius  is  ox ;  upon  the  surface  of  this  inner 
cylinder  lie  the  hehces  corresponding  to  xs^  ss,  in  Fig.  226. 

284.  Special  Case  of  the  Oblique  Helicoid.  In  Fig.  229,  the 
generatrix  DCJ^  cuts  the  axis  acutely  at  C;  the  helix  described  by 
the  point  C  therefore  coincides  with  the  axis  itself.  Let  c'm'  be 
one  half  the  pitch,  then  at  the  end  of  a  half  revolution  the  genera- 
trix will  have  the  position  e'-mlf  ^  which  intersects  the  first  position 
in  E\  the  path  of  this  point,  then,  is  the  first  of  the  helices  in 
which  the  generated  surface  intersects  itself.  Set  up  7}i  r'  equal  to 
the  pitch,  then  at  the  end  of  a  revolution  and  a  half  the  generatrix 
will  have  the  position  r's' ^  which  cuts  d'e'  produced  in  8\  whose 
path  s'xm'  is  the  second  of  the  series  of  helices,  as  explained  in 
(281). 

The  visible  contour,  as  in  all  other  cases,  is  tangent  to  the  pro- 
jections of  all  lines  of  the  surface  which  it  intersects  (except  those 
whose  rectilinear  tangents  at  these  points  of  intersection  are  per- 
pendicular to  the  plane  of  projection);  thus,  it  is  tangent  to  the 
helix  d'c'e\  and  at  V  and  ^',  to  the  path  of  the  point  K:  again, 
uu^  ww^  are  the  horizontal  projections  of  two  elements,  and  the 
contour  is  tangent  at  u'  and  w'  to  their  vertical  projections.  It  is 
also  obviously  tangent  to  the  axis  at  a\  the  middle  point  of  c'm' ; 
this  therefore  is  the  vertex  of  the  curve,  which  is  symmetrical  with 
respect  to  the  horizontal  line  through  a' ^  e' .  The  contour  line 
through  o\  when  produced,  as  o'q'^  will  also  be  tangent  to  the 
helix  mix's' ;  it  is  evidently  asymptotic  to  d'c's'^  and  closely  resem- 
bles the  hyperbola  having  the  same  vertex  and  asymptotes ;  but  the 
latter,  as  shown  by  the  broken  line  o'jp' ^  lies  within  the  curve  under 
consideration. 


DESCKlillVE   GEOMEIKY. 


1^7 


198  DESCRIPTIVE    GEOMETRY. 

285.  The  horizontal  trace  of  this  helicoid  is  an  Archimedean 
spiral.  If  c'd'  be  supposed  to  rotate  without  advaneing,  it  will 
after  half  a  revolution  occupy  the  position  c'y' ;  Imt  it  does  mean- 
time  actually  advance  to  the  position  me'  parallel  to  cy\  and  when 
produced  pierces  H  at  6^ :  and  since  c'm'  =  2o'c',  we  will  have 
y'g' —  'Ic'y' ^— d'y' ^  — de.  Since  the  revolution  and  the  axial 
advance  c:re  both  uniform,  the  trace  is,  as  stated,  an  equable  spiral, 
of  which  y'g\  or  eg^  is  the  radial  expansion  in  a  half  revolution. 
This  should  be  carefully  constructed,  by  dividing  eg  and  the  semi- 
circumference  dwe  into  proportional  parts,  and  setting  oft'  the  dis- 
tances cl,  c2,  etc.,  on  the  corresponding  radiants:  by  this  means, 
the  horizontal  traces  of  any  given  elements  can  be  more  accurately 
located  than  in  any  other  way. 

If  the  surface  be  limited  by  the  second  helix  of  intersection. 
m'x's\  a  section  by  a  transverse  plane  through y**'  will  be  bounded 
by  two  synmietrical  curves,  of  which  one  is  the  portion  dzg  of  this 
trace.  If  the  helicoid  be  limited  by  the  iirst  helix  d'ce'f\  the 
section  by  the  same  plane,  or  by  H  itself,  will  be  the  shaded  loop 
of  the  spiral :  the  meridian  section  by  a  plane  parallel  to  V  in  this 
case  consisting  merely  of  a  series  of  isosceles  triangles  whose  bases 
coincide  with  the  axis,  as  shown  on  a  reduced  scale  in  the  small 
detached  diagram. 

286.  Practical  Application.  This  helicoid,  as  previously  stated, 
forms  the  surface  of  the  Y-threaded  screw.  It  is  quite  obvious 
that  such  a  screw^  may  be  formed  by  taking,  in  Fig.  229,  the  helix 
m'x's  for  the  crest,  and  the  helix  d'c'e'  for  the  root,  of  the  thread ; 
both  sides  of  which  will  then  be  formed  of  portions  of  the  same 
helicoid.  But  it  is  not  essential  that  this  should  be  so ;  and  in  the 
majority  of  practical  cases  the  opposite  sides  of  the  thread  are  por- 
tions of  two  different  helicoids.  Thus  in  Fig.  230,  the  screw  may 
be  regarded  as  being  formed  by  winding  a  bar  of  flexible  metal, 
having  the  triangular  section  e7no^  around  the  central  core;  the 
pitch  trin  being  equal  to  eo.  Prolonging  me^  quo,  to  cut  the  axis 
in  c  and  d^  it  will  be  seen  that  if  cd  be  1^  times  the  pitch,  it  will 
correspond  to  c'r'  of  the  preceding  figure,  md  being  then  simply 
another  position  of  mc.  This,  however,  is  not  the  case ;  we  have 
here  two  independent  lines  each  generating  the  helicoidal  surface 


DESCRIPTIVE   GEOMETRY. 


199 


of  one  side  of  the  thread.  These  lines  are  usually,  as  in  this  illus- 
tration, equally  inclined  to  the  axis ;  which,  nevertheless,  is  by  no 
means  essential,  since  the  section  emo  of  the  flexible  bar  might 
have  been   a   right-angled,  or   a   scalene,  instead   of   an  isosceles 


Fig.  231 


triangle;  and  such  non-symmetrical  screw-threads  are  sometimes 
used  in  practice. 

If  the  pitch  be  doubled,  it  is  clear  that  another  thread  may  be 
formed  in  the  intervening  space ;  if  it  be  trebled,  two  threads  can 
be  added ;   and  so  on  indefinitely. 

For  the  sake  of  uniformity  the  helices  have  been  drawn  right- 


200  DESCRIPTIVE    GEOMETRY. 

handed  throughout ;  it  need  hardly  be  stated  that  they  might  have 
been  made  left-handed  without  in  any  way  affecting  the  argument : 
and  in  practice  they  often  are,  in  the  construction  of  ordinary 
screws  as  well  as  of  screw-propellers. 

287.  Helicoids  of  Yarying  Pitch.  If  in  Fig.  231  the  line  DC, 
which  is  perpendicular  to  die  vertical  axis,  were  to  rise  in  a  half 
revolution  to  the  positiony/',-  rotating  and  advancing  uniformly, 
it  would  generate  a  right  helicoid,  the  point  D  tracing  the  helix 
DGF.  Now  let  the  line  rotate  uniformly,  in  contact  with  this 
helix  and  with  the  axis,  the  point  C  at  the  same  time  advancing 
uniformly,  but  less  rapidly  than  before,  so  that  in  a  half  revolution 
it  reaches  the  altitude  ef  instead  of  l\  By  this  motion  a  different 
surface  w^ill  be  generated,  which  is  a  helicoid  of  varying  pitch : 
since  e'f  is  longer  than  c'd',  it  is  evident  that  the  point  Z>  does 
not  describe  the  helix  DGF^  but  since  (7  remains  in  contact  with 
the  axis,  there  is  a  sliding  of  the  generatrix  upon  the  guide  helix 
during  the  motion  supposed. 

Now  draw  the  inner  cylinder,  with  any  radius  oc  at  pleasure ; 
the  generatrix  in  its  first  position  pierces  this  cylinder  at  0^  and 
after  a  half  revolution,  pierces  it  at  P. 

Set  off  the  arc  dg^  say  one-third  of  the  semi- circumference,  and 
araw  gc^  the  horizontal  projection  of  the  generatrix  in  an  inter- 
mediate position ;  project  g  to  g'  on  the  outer  helix ;  the  altitude 
h'g'  will  be  one-third  of  s'f :  set  up  c'ti  =  -J  c'e\  then  g'h'  is  the 
vertical  projection  of  this  line,  which  pierces  the  inner  cylinder  at 

JV, 

It  is  obvious  that  the  altitude  m'n'  will  be  one-third  of  r'j?^ ; 
and  since  the  same  argument  applies  to  any  other  position  of  the 
generatrix,  it  follows  that  the  inner  cylinder  cuts  the  helicoid  in  a 
true  helix  ONP,  and  will  do  so  whatever  its  radhis  may  be,  the 
pitch,  evidently,  varying  with  the  radius :  and  the  same  applies  to 
exterior  cylinders  as  well. 

288.  In  Fig.  232,  the  generatrix  DC  \w  its  first  position  lies  in 
H ;  let  DBF  be  the  guide  helix,  and  FF  the  position  of  the  gen- 
eratrix after  a  half  revolution ;  let  mc,  nc  be  the  horizontal  pro- 
jections of  intermediate  positions;  if  these  be  revolved  about 
the  axis  until  parallel  to  V,  their  outer  extremities  will  appear  as 


DESCRIPTIVE    GEOMETRY.  201 

m\  7i\  dividing  s'f  in  the  same  projjortion  in  wliicli  their  inner 
extremities  divide  c'e.  These  lines  will  therefore  if  prolonged 
pierce  H  in  the  same  point  0 ;  as  also  will  k'l  \  the  position  of  the 
generatrix  after  a  revolution  and  a  half — from  which  it  follows  that 
all  the  elements  of  this  surface  pierce  H  at  the  same  distance  from 
the  axis.  In  other  words  the  horizontal  trace  of  the  surface  con- 
sists of  the  circle  whose  radius  is  eo,  and  also  of  an  indefinite  right 
line  coincidino;  with  Z^6\  the  element  which  lies  in  the  horizontal 
plane. 

289.  The  transverse  section  by  any  other  plane  as  PP  will  be 
a  spiral,  wux,  of  which  the  points  u  and  w  are  obtained  directly, 
since  the  elements  g7i\  Vh\  in  the  vertical  projection  pierce  this 
plane  at  u'  ^  w' .  To  find  other  points,  draw  intermediate  elements, 
as  through  m  and  n  in  the  horizontal  projection ;  these  when  re- 
volved until  parallel  to  V  will  pass  through  in" ^  n'\  points  dividing 
h'h"  into  parts  proportional  to  the  divisions  of  g'V^  and  will  be  cut 
by  PP  at  points  whose  distances  from  the  axis  are  to  be  set  off  on 
cm^  en :  and  in  like  manner  any  number  of  points  may  be  deter- 
mined, and  the  curve  extended  in  either  direction.  This  curve  is 
peculiar  in  possessing  a  circular  as  well  as  a  rectilinear  asymptote ; 
if  continued  in  the  direction  uw^  it  will  pass  through  the  pole  c, 
and  then  again  expanding  at  a  decreasing  rate,  it  will  after  an  in- 
finite number  of  turns  be  tangent  internally  to  the  circle  oqt :  if 
continued  in  the  opposite  direction,  it  will  be  tangent  at  infinity  to 
a  line  parallel  to  ed,  and  lying  at  a  distance  from  it  equal  to  the 
circumference  of  the  circular  asymptote. 

290.  In  Fig.  233,  the  generatrix  PC  in  its  first  position  also 
lies  in  H,  but  at  a  distance  cic  from  the  axis ;  DKF  is  a  helix  traced 
on  the  cylinder  whose  radius  is  ud.  Now  let  DC  revolve  uniformly 
about  the  axis,  in  contact  with  this  helix,  the  point  C  at  the  same- 
time  moving  uniformly  along  the  element  of  contact  with  the  cyl- 
inder whose  radius  is  cu^  so  that  after  a  half  revolution  the  line 
occupies  the  position  FE.  By  reasoning  similar  to  that  used  in 
connection  with  Fig.  231,  it  can  be  shown  that  the  surface  thus 
generated  will  intersect  any  cylinder  having  the  same  axis  in  a  true 
helix;  this  form  of  the  helicoid  is,  then,  the  general  one,  of  whicl^ 
that  above  discussed  is  a  special  case. 


202 


DESCRIPTIVE   GEOMETRY. 


Fig.  233 


1  ^ 

p 

/   >~X    /      'A 

""'' 

L 

, 

\ 

\ "1 

~c "   y 

/     \    /  /\.   V 

\ 

\  //  1  A    1  ; 

(  If 

\ 

1        w     \      /  J  <^ 

'■■\ 

/// 

M 

'j^  yfCj 

i^- 

TV 

]■  ( 

^.Qi 

-==>_. 

r 

^1 

Fig.  234 


DESCRIPTIVE    GEOMETRY.  203 

Drawing  mj?,  nr^  tangent  to  tlie  circle  whose  radius  is  uc^  and 
revolving  the  elements  of  which  these  are  the  horizontal  projections 
nntil  thej  are  parallel  to  T,  it  mav  be  shown  as  in  Fig.  231  that  they 
will  then  pierce  H  in  the  same  point  O  :  showing  that  the  horizontal 
trace  of  this  surface  consists  of  the  circle  whose  radius  is  uo^  and 
also  of  a  right  line  coinciding  with  DC^  the  original  position  of  the 
generatrix. 

291.  Practical  Application.  AVhen  the  generatrix  intersects 
the  axis,  the  helicoid  of  varying  pitch  is  often  practically  employed, 
forming  the  acting  surfaces  of  screw-propellers  with  what  is  techni- 
cally called  ' '  radially  increasing  pitch ' ' ;  which  are  swept  up,  of 
course,  by  that  portion  only  of  the  generatrix  which  lies  on  one  side 
of  the  axis. 

To  aid  in  gaining  a  clear  idea  of  the  nature  of  this  surface, 
there  is  represented  in  Fig.  234  a  portion  of  it  generated  by  a  line 
of  definite  length  JS/Z;  the  pitch  at  that  distance  from  the  axis 
being  LP^  PE^  while  the  pitch  at  the  axis  itself  is  MC^  CD.  The 
generatrix  ML  being  perpendicular  to  the  axis,  will  after  successive 
revolutions  take  the  positions  CEP^  DFR^  etc.  ;  the  point  L  thus 
tracing  a  ^w<a^6- /-helical  curve  LIEK^  Iji^g  on  a  surface  of  revolu- 
tion whose  meridian  line  is  LEFG\  since,  as  shown  in  (288), 
7?Z>,  PC^  etc.,  when  produced  w^ill  meet  in  the  same  23oint  0  on 
the  prolongation  of  LM^  the  outline  of  this  surface  of  revolution  is 
tlie  waved  branch  of  a  conchoid,  of  w^hich  the  pole  is  0  and  the 
directrix  is  NN  the  axis  of  the  helicoid.  .  In  order  to  throw  the 
surface  into  stronger  relief,  a  concentric  cylinder  is  introduced, 
which,  as  before  shown,  cuts  it  in  a  true  helix,  of  which  portions 
are  visible. 

292.  From  this  it  is  clearly  seen  that  the  surface  is  divided  into 
two  symmetrical  parts  by  the  median  plane  LL^  which  contains 
the  element  perpendicular  to  the  axis.  In  the  immediate  neighbor- 
hood of  this  plane  the  surface  resembles  the  right  helicoid,  while  at 
sensible  distances  from  it  there  is  a  greater  similarity  to  the  oblique 
helicoid.  And  it  is  specially  to  be  noted,  that  the  generatrices  in- 
cline in  opposite  directions  on  the  two  sides  of  this  plane ;  so  that 
in  the  construction  of  a  propeller  it  does  not  suffice  to  giye  merely  the 
pitches  at  the  rim  and  the  hub  respectively ;   since,  while  the  form 


204  DESCRIPTIVE   GEOMETRY. 

of  the  surface  would  thus  be  definitely  fixed,  the  particular  pprt  of 
it  to  be  used  would  not  l)e  located :  it  is  necessary  therefore  to  give 
in  addition  the  precise  inclination  to  the  axis  of  some  specified  rec- 
tilinear element  of  the  proposed  blade. 

THE    CYLINDEOID. 

293.  The  Cjlindroid  is  a  warped  surface  with  a  plane  directer, 
and  is  derived  from  the  cylinder  in  a  manner  which  will  be  readily 
understood  by  the  aid  of  the  pictorial  representation,  Fig.  235.  On 
the  left  is  shown  the  half  of  a  circular  cylinder,  with  its  axis  in  II 
and  its  elements  parallel  to  V ;  and  this  semi-cylinder  is  cut  obliquely 
by  two  vertical  planes,  forming  the  sections  erd^  msn  :  the  elements 
cut  the  outlines  of  these  sections  in  the  corresponding  points  1  1 , 
2  2,  etc. 

Now,  the  section  erd  remaining  fixed,  let  the  other  section  be 
moved  upward  by  translation  in  its  own  plane  through  any  given 
distance;  the  diameter  mon  will  then,  as  shown  on  the  right,  be 
vertically  over  and  parallel  to  its  orginal  position,  here  represented 
by  m,6»,7i, :  moreover,  the  relative  positions  of  the  points  1,  2,  3, 
upon  the  arc  ns^  will  be  the  same  as  before. 

]S"ext,  joinmg  these  points  with  the  corresponding  points  1,  2, 
3,  upon  the  arc  di\  the  new  lines  11,  2  2,  etc.,  will  be  the  rectilinear 
elements  of  the  surface  under  consideration.  By  construction  these 
elements  are  parallel  to  T,  which  therefore  is  the  plane  directer  in 
the  case  here  illustrated. 

The  method  of  representing  this  surface  in  projection  is  too 
obvious  from  the  above  to  require  further  explanation ;  nor  does 
the  surface  itself  possess  any  remarkable  features,  with  the  excep- 
tion that  the  limiting  tangent  planes,  as  LL  for  instance,  are  tan- 
gent all  along  the  elements  which  they  contain. 

294.  Practical  Application.  The  cylindroid  may  be  used  to 
form  the  roof  of  a  transverse  gallery  connecting  two  parallel  arched 
passages  on  different  levels. 

The  floor  of  such  a  gallery,  if  constructed  on  the  same  principle, 
will  also  be  a  warped  surface ;  it  is  evident  that  in  the  right-hand 
figure,  mn^  de^  are  the  directrices,  and  nd^  me^  are  two  elements, 


DESCRIPTIVE   GEOMETRY 


205 


206  DESCRIPTIVE    GEOMETRY. 

of  a  hyperbolic  paraboloid,  wliick  lias  H  for  one  plane  directer  and 
'*''  for  the  other. 

Ill  the  preceding  illustration,  the  circular  cylinder  was  selected 
merely  for  convenience ;  it  is  clear  that  a  similar  process  may  be 
employed,  whether  the  roof  of  the  original  arch  be  circular,  ellipti- 
cal, or  of  any  other  section. 

THE    cow's    HORN. 

295.  This  is  a  warped  surface  liaying  three  directrices,  viz.,  a 
right  line,  and  two  circles  in  parallel  planes  :  a  plane  perpendicular 
to  the  latter  contains  the  centres  of  both  circles  and  also  the  recti- 
linear generatrix. 

In  Fig.  236,  the  circles  onrn^  hsd,  are  of  equal  diameters  and 
lie  in  vertical  planes,  to  which  the  rectilinear  directrix  2)0  is  per- 
pendicular ;  also,  the  centres,  c  and  e,  are  on  opposite  sides  of  po 
and  equidistant  from  it.  Under  these  conditions  tlie  surface  is 
syjiimetrically  divided  by  the  vertical  plane  through  ^>>6> ;  and  has  a 
practical  application  in  the  construction  of  the  warped  arch. 

Any  plane  through  po,  evidently,  will  cut  the  planes  of  the 
circles  in  parallel  lines,  as  ok,  pi,  thus  determining  an  element  Ik 
of  the  surface.  The  element  mb  cuts  po  in  a? ;  gk  cuts  it  in  w, 
farther  from  o ;  Ik  would  cut  it  in  a  point  still  more  remote,  and 
so  on,  until,  when  the  cutting  plane  becomes  vertical,  pr  and  os 
being  equal  under  the  assumed  conditions,  rs  is  parallel  to  po. 
The  elements  beyond  rs  will  then  intersect  op  on  the  opposite  side 
of  the  vertical  plane ;  dn,  evidently,  cutting  it  at  a  point  y,  mak- 
ingpy  equal  to  ox. 

in  Fig.  237,  none  of  the  above  special  conditions  are  imposed ; 
the  two  circles  are  of  different  diameters,  the  directrix  po  is  not 
perpendicular  to  their  planes,  and  the  centres  c  and  e  lie  on  the 
same  side  of  po,  but  at  unequal  distances  from  it ;  moreover  these 
distances  are  so  chosen  that  the  points  p  and  o  do  not  divide  tlie 
radii  en,  ed,  in  the  same  proportion.  This  figure,  then,  represents 
a  general  case  of  the  surface,  of  which  the  warped  arch  is  only  a 
special  form.  All  the  elements  now  cut  the  rectilinear  dii-ectrix  in 
ironfc  of  the  vertical  plane,  and  at  finite  distances,  since  no  one  of 
them  is  parallel  to  it. 


DESCRIPTIVE    GEOMETRY.  20? 

296.  The  obvious  nse  of  sucli  surfaces  is  in  forming  the  roofs 
of  arched  passages ;  which  naturally  leads  to  the  selection  of  circles, 
ellipses,  or  other  symmetrical  curves,  lying  in  parallel  planes,  for 
the  curved  directrices.  The  construction  of  the  roof  requires  the 
use  of  only  one-half  of  each  of  these  curves,  which  accordingly  is 
all  that  is  i^hown  in  the  figures :  if  we  suppose  the  other  half  to  be 
added,  It  is  evident  that  the  plane  containing  the  centres  of  these 
curves  and  the  rectilinear  directrix,  will  divide  the  complete  sur- 
face symmetrically,  and  in  general  no  other  plane  will  do  so.  The 
above  pictorial  representations  show  not  only  the  nature  of  the  sur- 
face, but  the  method  of  determining  its  elements,  more  clearly  than 
would  its  projections,  which  can  readily  be  drawn  without  further 
explanation.  Since  no  plane  can  be  tangent  to  such  a  surface  along 
an  element,  the  visible  contour  will  in  all  cases  be  a  curve,  though 
sometimes  a  very  fiat  one. 

Substantially  the  same  method  would  be  employed  were  the 
curved  directrices  in  planes  not  parallel  to  each  other,  not  similar 
to  each  other,  or  even  were  they  of  double  curvature ;  and  indeed 
it  may  be  said  that  the  Cow's  Horn  is  only  a  special  variety  of  a 
general  class  of  warped  surfaces,  having  one  rectilinear  directrix  and 
two  curved  ones  of  any  kind  whatever. 

WARPED    SURFACES GENERAL    FORMS. 

297.  In  addition  to  the  preceding,  warped  surfaces  having  no 
specific  names  are  sometimes  met  with  in  practical  operations. 
These  must  necessarily  be  determined  either  by  two  curved  direc- 
trices and  a  plane  director,  or  by  three  curvilinear  directrices ;  and 
in  representing  them,  three  problems  may  arise.  If  there  be  a 
plane  directer,  it  may  be  required  to  draw  an  element  either 
parallel  to  a  given  line  therein,  or  through  a  given  point  on  one  of 
the  directrices ;  if  there  be  none,  it  may  be  required  to  draw  an 
element  through  a  given  point  upon  either  directrix.  We  will 
consider  these  problems  in  the  order  given. 

298.  I.  In  Fig.  238,  let  CD,  EF,  be  the  directrices  of  a 
warped  surface ;  it  is  required  to  draw  an  element  parallel  to  the 
line  jollying  in  the  plane  directer  tTt' . 

Analysis.     Pass  through  either  directrix  a  cylinder  whose  ele^ 


208 


DESCKIPTIVE    GEOMETRY. 


ments  are  parallel  to  the  given  line.  The  other  directrix  will 
pierce  this  cylinder  in  one  or  more  points,  through  either  of  which 
an  element  of  the  surface  may  be  drawn  parallel  to  the  given  line. 

Construction.  Through  any  points  G^  11^  K^  etc. ,  on  (7i>,  draw 
parallels  to  MN\  these  lines  pierce  the  vertical  projecting  cylinder 
of  EF  in  the  points  7?,  0^  Z,  etc. ,  thus  determining  a  curve  verti- 
cally projected  in  €'f\  and  horizontally  projected  in  e^f^.  This 
curve  is  the  intersection  of  the  two  cylinders,  and  cuts  EF  in  the 
point  Z7,  through  which  is  drawn  the  required  element  ZZZ,  paral- 
lel to  M]S\ 

299.  II.  In  Fig.  239,  let  OD,  EF,  be  the  directrices,  tTt'  the 


Fig.  238 


plane  directer ;  it  is  required  to  draw  an  element  of  the  warped 
surface,  through  the  point  O  on  CD, 

Analysis.  Pass  through  the  given  point  a  plane  parallel  to  the 
plane  directer;  it  will  cut  the  other  directrix  in  a  point  of  the 
required  element. 

Construction.  Assume  in  tTt'  any  point  P  and  also  any  line 
JO^,  and  join  P  by  right  lines  to  any  points  B,  /S,  etc. ,  upon  MJV, 
Through  O  draw  parallels  to  PP,  PS,  etc.  ;  this  series  of  lines 
determines  a  plane  parallel  to  tTt'.  This  parallel  plane  cuts  the 
horizontal  projecting  cylinder  of  EF  in  a  curve  whose  projections? 
are  ef,  e^f^ ;  and  this  curve  cuts  EF  in  X,  thus  determining  OX^ 
the  element  required. 


DESCRIPTIVE   GEOMETRY. 


209 


300.  III.  In  Fig.  240,  MN,  CD,  EF,  are  the  directrices  of  a 
warped  surface;  it  is  required  to  draw  a  rectilinear  element 
through  the  point  O  on  MN. 


Fig.  239 

Analysis.  Pass  through  either  of  the  other  directrices  a  cone 
of  which  the  given  point  is  the  vertex.  The  third  directrix  will 
pierce  this  cone  in  one  or  more  points,  through  either  of  w^hich 
and  the  given  point  an  element  of  the  surface  may  be  drawn. 

/A 


FiQ.  240 


Construction.  Through  any  points  G,  R,  K,  etc. ,  on  CD,  draw 
lines  from  O ;  these  are  elements  of  the  cone,  and  pierce  the  hori- 
zontal projecting  cylinder  of  EF  in  the  points  H,  jS,  Z,  etc  :  thus 


210  DESCRIPTIVE    GEOMETRY. 

determining  a  curve,  of  wliicli  the  horizontal  projection  is  ef  and 
the  vertical  is  e„f^.  This  curve  is  the  intersection  of  the  cone 
with  the  projecting  cylinder,  and  cuts  EF  in  X,  a  point  of  the  re- 
quired element  OX. 

PLANES    TANGENT    TO    WARPED    SURFACES. 

303..  A  plane  tangent  to  any  warped  surface  at  a  given  point 
may  be  constnicted  by  the  general  rule  of  (140),  viz.  :  Draw 
through  the  point  any  two  intersecting  lines  of  the  surface,  and  at 
the  point  a  tangent  to  each  line ;  these  tangents  determine  the  re- 
quired plane. 

The  rectilinear  element  through  the  given  point  is  one  line  of 
the  tangent  plane,  in  all  cases ;  and  if  the  surface  be  doubly  ruled, 
the  plane  is  at  once  determined  by  drawing  through  the  point  an 
element  of  each  generation  (142) ;  as  has  already  been  illustrated 
in  the  cases  of  the  hyperbolic  paraboloid  (236),  the  hyperboloid  of 
revolution  (250),  and  the  elliptical  hyperboloid  (260). 

If  there  be  only  one  set  of  rectilinear  elements,  that  curve  of 
the  surface  should  be  selected  to  which  the  tangent  can  most  read- 
ily be  drawn :  the  helicoid  affords  a  good  illustration,  since  the 
tangent  to  the  helix  is  easily  determined. 

302.  Problem  1.  To  draw  a  j^lane  tangent  to  an  oblique  heli- 
coid at  a  given  point. 

Construction.  In  Fig.  241,  let  2>(7,  parallel  to  V,  be  the  gen- 
eratrix, and  DFE  the  helix  traced  by  D :  to  draw  a  plane  tangent 
to  the  surface  at  the  point  P  upon  this  helix. 

Since  the  generatrix  cuts  the  axis,  set  up  c'g'  =  p'y\  then  g'p' 
is  the  vertical  projection  of  the  element  through  P,  whose  traces 
are  0  and  S.  Since  tho  given  helix  pierces  H  at  Z>,  makej?/',  per- 
pendicular to  cp^  equal  to  the  arcj9(i;  then  rpm  is  the  horizontal, 
and  r'p'w!  is  the  vertical,  projection  of  the  tangent  at  P  to  the 
helix.  It  is  the  horizontal  trace  of  this  tangent ;  therefore  tovT  is 
the  horizontal,  and  Ts't'  is  the  vertical,  trace  of  the  required  tan- 
gent plane. 

Note.  The  horizontal  trace  of  this  surface  is  the  Archimedean 
spiral  hodl.^  constructed  as  in  Fig.  229 ;  and  it  is  to  be  particularly 
observed  that  tT  is  not  tangent  to  this  trace,  because  the  plane  is 


DESCRIPTIVE   GEOMETRY. 


211 


tangent  to  the  surface  only  at  the  point  P^  and  not  along  an  ele- 
ment.    The  horizontal  trace  of  a  plane  tangent  to  the  helicoid  at 


Fig.  241a 


2>,  or  any  other  point  in  H,  of  course  would  be  tangent  to  this 
spiral. 


^18  DESCRIPTIVE   GEOMETRY. 

303.  Problem  2.  To  find  the  jj^int  of  tangency  hetween  a 
given  ohliqii^e  helicoid  and  a  jplane  containing  a  given  element 
thereof. 

Analysis.  Since  the  plane  is  not  parallel  to  the  other  elements, 
it  will  cut  each  of  them  in  a  point ;  tlie  curve  passing  through  the 
points  thus  found  will  cut  the  given  element  in  tlie  required  point. 

Construction.  In  Fig.  242,  let  DC  parallel  to  V,  be  the  gen- 
eratrix of  the  helicoid,  whose  pitch  is  also  given.  Let  GO  be  the 
given  element  of  the  surface  thus  determined,  and  tTt'  the  given 
plane. 

Having  drawn  the  horizontal  trace  of  the  surface,  hodl^  as 
before,  draw  <?m,  c?i,  etc.,  the  horizontal  projections  of  elements 
on  each  side  of  GO.  The  points  in  which  these  elements  pierce 
tTt'  can  be  most  readily  found  by  means  of  a  supplementary  pro- 
jection on  a  plane  zu'z'\^  perpendicular  to  tT.  In  that  projection, 
the  intersection  of  this  plane  with  the  given  plane  appears  as  tlie 
line  2,s/,  the  axis  as  c,<?,  perpendicular  to  IIH.,  and  the  points 
^7^,  71,  /*,  are  projected  at  7n^ ,  n^ ,  'i\ ,  respectively.  The  rate  of 
axial  advance  being  known,  the  supplementary  projections  of  the 
elements  through  m,,  t?,,  etc.,  are  readily  determined;  they  cut  s.s/ 
at  ^«?l ,  ^, ,  y, ,  whence  they  are  projected  back  to  %o  on  cm,  e  on  en., 
etc.,  thus  determining  the  curve  wf^  which  cuts  co  at^,  the  hori- 
zontal projection  of  the  point  sought. 

304.  Problem  3.  To  draw  a  jplane  tangent  to  an  oblique  heli- 
coid^ and  perjpendicular  to  a  given  right  line. 

Analysis.  Construct  the  cone  directer  of  the  surface,  and  draw 
through  its  vei'tex  an  auxiliary  plane  perpendicular  to  the  given 
line.  If  the  problem  be  possible,  this  plane  will  in  general  cut 
from  the  ronfe  two  elements;  these  are  respectively  parallel  to 
those  elements  of  the  helicoid,  through  either  of  which  a  plane 
may  be  drawn  parallel  to  the  auxiliary  plane  and  tangent  at  some 
point  to  the  surface.  The  point  of  contact  is  found  as  in  the  pre- 
ceding problem. 

Construction.  In  Fig.  241a,  6^  is  the  vertex  and  DYXh  the 
base  of  the  cone  directer,  LL  is  the  given  line,  sSs  is  the  auxiliary 
plane  perpendicular  to  it,  and  cp^  en,  are  the  horizontal  projections 
of  the  two  elements :  had  the  plane  been  tangent  to  the  cone,  it  is 


DESCRIPTIVE    GEOMETRY. 


213 


214  DESCRIPTIVE    GEOMETRY. 

evident  that  onlj  tlie  elements  of  the  helicoid  parallel  to  the  line  of 
contact,  could  have  planes  drawn  through  them  which  would  satisfy 
the  assigned  condition. 

Note.  The  cone  directer  here  drawn,  is  that  of  the  helicoid 
represented  in  Figs.  241  and  242 ;  in  each  of  which  also  the  tan- 
gent plane  is  parallel  to  s/Ss'  of  this  diagram,  and  contains  the  same 
element  parallel  to  CP ;  which  facilitates  a  comparison  of  the  con- 
structions. 

305.  The  same  general  method,  evidently,  may  be  employed 
in  dealing  with  any  warped  surface  having  a  cone  directer,  in  re- 
gard to  which  the  preceding  problem  may  be  proposed.  In  rela- 
tion to  the  hyperboloid  of  revolution  and  the  elHptical  hyperboloid, 
it  is  to  be  noted  that  since  they  are  doubly  ruled,  the  two  elements 
parallel  to  those  cut  from  the  cone  directer  at  once  determine  the 
tangent  plane,  and  their  intersection  determines  the  point  of  con- 
tact. If  the  auxiliary  plane  be  tangent  to  the  cone  directer, 
the  two  elements  of  the  surface  parallel  to  the  line  of  contact, 
will  in  the  transverse  section  be  tangent  to  the  gorge  curve 
on  opposite  sides,  and  the  point  of  contact  will  be  infinitely 
remote. 

306.  Problem  4.  To  draw  a  plane  tangent  to  a  liyperholiG 
jparaholoid^  and  perpendicular  to  a  given  rigid  line. 

Argument.  This  may  be  accomplished  by  the  following  series 
of  operations,  viz.  : 

1.  Draw  the  plane  directors  and  find  their  line  of  intersection. 

2.  Draw  a  plane  perpendicular  to  the  gi^en  line. 

3.  Find  the  intersection  of  this  plane  with  each  plane  directer. 

4.  Find  an  element  of  the  surface  parallel  to  each  of  these 
lines  of  intersection  (226). 

These  elements  will  be  of  diiferent  generations,  w^ll  determine 
a  plane  perpendicular  to  the  given  line,  and  will  intersect  in  the 
point  of  contact. 

307.  Problem  5.  To  draw  a  plane  tangent  to  any  icarped 
surface^  and  perpendicular  to  a  given  rigid  line. 

Preliminary,  In  Fig.  243,  let  ZZ  be  the  given  line,  6'aS^' the 
plane  perpendicular  to  it.  Through  any  point  P  in  this  plane, 
draw  JO^  parallel  to  the  horizontal  trace,  and  7** 6^  parallel  to  the 


DESCRIPTIVE    GEOMETRY.  215 

vertical  trace.      The  following  is  then  applicable  to  any  warped 
surface  whatever. 

Argument.  1.  Draw  a  series  of  sections  of  the  given  surface  by 
horizontal  planes;  draw  a  tangent  to  each  section,  parallel  to  MN^ 
and  find  the  point  of  contact.  The  line  joining  these  points  is  the 
curve  of  contact  between  tlie  warped  surface  and  a  cylinder  whose 
elements  are  parallel  to  MN. 

2.  Draw  a  series  of  sections  of  the  given  surface  by  planes 
parallel  to  V;  draw  a  tangent  to  each  section  parallel  to  liO,  find 
the  point  of  contact,  and  draw  a  curve  through  the  points  thus 
found ;  this  is  the  line  of  contact  with  a  second  cylinder  whose  ele- 
ments are  parallel  to  BO. 

3.  The  two  curves  thus  determined  wdll  intersect  in  the  re- 
quired point  of  contact  between  the  given  surface  and  a  plane  per- 
pendicular to  the  given  line :  which  plane  will  of  course  contain 
the  rectilinear  elements  of  the  surface  which  pass  through  the 
point. 

Note,  The  lines  MW^  RO^  have  in  the  above  argument  been 
mrxle  parallel  to  H  and  V  respectively,  for  convenience  only ;  it  is 
clear  that  any  other  lines  in  sSs'  might  have  been  used  as  well,  the 
elements  of  the  two  tangent  cylinders  being  drawn  parallel  to  them  : 
but  in  general  the  execution  would  be  more  laborious. 

308.  Problem  6.  To  draw  a  plane  tangent  to  any  vmrped 
surface.^  through  a  given  right  line. 

Argument.  Let  the  given  line  be  produced  until  it  pierces  the 
surface ;  then  the  rectilinear  element  through  the  point  of  penetra- 
tion, and  the  given  line  itself,  determine  a  plane  which  in  general 
wdll  be  tangent  to  the  surface  at  some  point  of  the  element. 

If  two  elements  pass  through  the  point,  each  will  determine  a 
tangent  plane;  so  again,  if  the  given  line  pierce  the  surface  in 
more  than  one  point,  there  will  be  more  than  one  tangent  plane. 
If  the  given  line  be  parallel  to  a  rectilinear  element,  the  point  of 
penetration  will  be  infinitely  remote,  and  the  tangent  plane  is  de- 
termined by  that  element  and  the  given  line  itself.  The  point  of 
tangency  is,  in  all  cases,  the  intersection  of  the  rectilinear  element 
with  the  curve,  if  there  be  one,  cut  from  the  warped  surface  by 
the  tangent  plane. 


216  DESCllIPTIVE    GEOMETRY. 

There  are  cases  in  wliicli  there  is  no  such  curve — for  instance, 
the  given  line  may  He  in  a  plane  tangent  to  a  conoid  or  cylindroid 
along  an  element; — or,  the  plane  determined  by  the  given  line  and 
an  element  of  a  hyperbolic  paraboloid,  may  be  parallel  to  a  plane 
directer. 

309.  The  problems  of  passing  a  j^lane  tangent  to  a  warped  sur- 
face, and  either  parallel  to  a  given  right  line,  or  through  a  given 
point  without  the  surface,  are  indeterminate. 

In  the  first  case,  by  making  a  sei-ies  of  sections  of  tlie  surface 
by  planes  parallel  to  the  given  line,  and  drawing  a  tangent  to  each, 
also  parallel  to  it,  a  cylinder  may  be  constructed,  tangent  to  the 
surface.      Any  plane  tangent  to  this  cylinder  satisfies  the  conditions. 

In  the  second  case,  a  series  of  sections  of  the  warped  surface 
being  made  by  planes  containing  the  given  point,  let  a  tangent  be 
drawn  to  each  through  that  point.  These  tangents  are  elements  of 
a  cone  tangent  to  the  given  surface,  whose  vertex  is  the  given 
point :   and  any  plane  tangent  to  this  cone  satisfies  the  conditions. 

310.  Use  of  Auxiliary  Surfaces.  The  operation  of  drawing  a 
plane  tangent  to  a  w^arped  surface  may  sometimes  be  facilitated  by 
the  use  of  another  warped  surface.  It  w^ill  presently  appear  that, 
as  stated  in  (144),  one  such  surface  may  be  tangent  to  another  one 
all  along  an  element.  In  that  case,  any  plane  tangent  to  either  is 
tangent  to  both,  if  the  jjoint  of  contact  lies  on  the  common  ele- 
ment ;  and  as  has  already  appeared,  it  may  be  much  easier  to  draw 
a  plane  tangent  to  the  one  than  to  the  other. 

TANGENCY  OF  WARPED  SURFACES. 

311.  Two  warped  surfaces  are  tangent  to  each  other,  like  any 
others,  wdien  they  have  at  any  point  a  common  tangent  plane.  In 
order  that  they  may  be  tangent  all  along  a  common  element,  they 
nuist  have  a  common  tangent  plane  at  every  point  thereof.  And 
this  will  be  the  case,  if  that  condition  be  satisfied  for  any  three 
points  of  the  given  element. 

For,  in  Fig.  244,  let  LL  be  an  element  common  to  two  given 
w^arped  surfaces,  which  have  a  common  tangent  plane  at  each  of 
the  three  points  A^  B^  C.  Any  intersecting  planes  passed  through 
theso  points  will  cut  from  one  surface  the  three  curves  i>,  E^  F^ 


DESCRIPTIVE    GEOMETRY.  217 

from  tlie  other  tlie  curves  G^  11^  Z,  and  from  the  tangent  planes 
tlie  riglit  hnes  R^  S,  T. 

The  curves  D^  G,  being  tangent  to  each  otlier,  have  two  con- 
secutive pomts  in  common ;  and  the  same  is  tnie  of  the  other 
pau-s,  E^  H,  and  F^  I.  Consequently,  if  LL  be  moved  eitlier  upon 
D^  E^  F^  or  upon  G^  H^  /,  as  directrices,  into  its  consecutive  posi- 
tion, it  will  lie  in  both  surfaces*  which,  therefore,  have  two  con- 
secutive rectilinear  elements  in  common.  Any  plane  cutting 
tliese,  evidently,  will  cut  from  the  surfaces  two  lines  which  have 
two  consecutive  points  in  common,  or  in  other  words  are  tangent 
to  each  other :  tlie  two  surfaces,  then,  are  tangent  all  along  LL. 

312.  If  the  two  surfaces  have  a  common  plane   directer,  and 


common  tangent  planes  at  two  points  upon  a  common  element, 
they  will  be  tangent  all  along  that  element.  For  in  this  case  the 
motion  of  L^L  in  Fig.  244  would  be  completely  determined  by  two 
of  the  pairs  of  curves  there  shown ;  and  the  argument  is  otherwise 
the  same  as  above. 

313.  If  in  the  same  figure  the  generatrix  LL  be  moved  upon 
the  tangent  lines  B^  /S,  T,  as  directrices,  instead  of  upon  either  set 
of  curves,  it  will  generate  a  third  warped  surface,  tangent^  all,  along 
the  element,  to  both  the  others:  this  surface  having  three  recti- 
linear directrices,  must  be  eitlier  a  hyperbolic  paraboloid,  or  a 
warped  hyperboloid.  If  the  three  intersecting  planes  passed 
through  J.,  B^  and  (7  are  parallel,  then  R^  S,  and  7^  are  all  parallel 
to  one  plane,  and  the  surface  is  a  hyperbolic  paraboloid ;  if  they 
are  not,  the  surface  will  be  either  a  circular  or  an  elliptical  hyper- 
boloid.    The  relations   and  directions  of  the  intersecting   planes 


218  DESCRIPTIVE    GEOMETRY, 

being  entirely  arbitrary,  any  number  of  sets  of  planes,  parallel  or 
not,  may  be  drawn  tlirougli  those  points ;  consequently,  any  num- 
ber of  hyperbolic  paraboloids,  and  any  immber  of  warped  hyperbo- 
loids,  may  be  constructed,  all  of  which  shall  be  tangent  along  the 
element  LL  to  both  the  given  surfaces. 

314.  Since  the  directions  of  the  tangents  7?,  8^  T,  are  deter- 
mined by  either  of  the  two  given  surfaces  independently  of  the 
other,  it  might  seem  that  the  above  would  be  true  of  any  element 
of  any  warped  surface.  But  there  are  exceptions.  It  has  been 
seen  that  a  plane  may  be  tangent  all  along  certain  elements  of  a 
warped  surface,  as  in  the  cases  of  the  cylindroid  and  some  forms  of 
the  conoid.  To  such  a  surface,  evidently,  no  other  warped  surface 
can  be  tangent  along  those  elements,  except  one  which  possesses 
the  same  peculiarity,  and  this  is  conspicuously  not  the  case  with  the 
hyperboloids :  single-curved  surfaces,  however,  may  be  so,  of  any 
kind  and  of  any  number. 

The  normals  at  all  points  of  such  an  element,  being  perpendicu- 
lar to  one  plane,  are  parallel  to  each  other,  and  thus  determine  a 
plane  normal  to  the  surface  all  along  the  element. 

315.  Tlie  normals  to  a  warped  surface  at  various  points  of  a 
given  element  are  not  in  general  tlius  perpendicular  to  any  one 
plane.  But  if  they  are  not,  tlien,  whatever  tlie  nature  of  the  given 
surface,  these  normals  are  elements  of  a  rectangular  hyperbolic 
paraboloid. 

This  may  be  shown  as  follows :  In  Fig.  245,  let  Z>,  E^  F^  be  the 
curves  cut  from  a  warped  surface  by  planes  perpendicular  to  the 
element  LL  at  the  points  A^  B^  C\  let  JT,  Y^  Z  be  the  normals  at 
those  points,  which  will  be  perpendicular  to  the  tangents  /^,  S^  T 
lying  in  the  same  intersecting  planes.  If  LL  be  moved  upon  the 
tangents  to  any  new  position  MN^  it  will,  as  has  already  been  seen, 
generate  a  hyperbolic  paraboloid ;  of  which  one  plane  director  will 
be  any  plane  F  perpendicular  to  LL^  and  the  other  will  be  any 
plane  H  parallel  to  both  MN  and  LL^  and  consequently  perpeii- 
dicular  to  Y.  This  tangent  paraboloid,  then,  is  rectangular ;  and 
if  it  be  revolved  about  LL  through  an  angle  of  90°,  the  elements 
^,  S^  T  will  coincide  with  the  normals  X,  Y^  Z,  the  line  MI^ 
taking  the  position  TJW.     The  paral)oloid  now  has  for  one  set  of 


DESCRIPTIVE   GEOMETRY. 


219 


elements  the  series  of  normals,  and  for  one  plane  director  the  plane 
V  to  which  they  are  all  parallel ;  the  other  plane  directer  will  be 
any  plane  P  parallel  to  both  UW  and  ZZ,  and  therefore  perpen- 
dicular to  V:  and  since  the  revolution  was  through  an  angle  of 
90°,  this  plane  P  will  also  be  perpendicular  to  ZT. 

Note.  In  this  illustration,  the  conditions  have  for  the  sake  of 
clearness  been  so  chosen  that  the  plane  directors  F,  Zf,  P,  are 
respectively  parallel  to  the  vertical,  horizontal,  and  profile  planes; 
and  for  further  elucidation,  the  positions  of  the  tangents  i?,  S,  T, 


N 

M 

c 

B 

T—sN 

S        \ 

\ 

A 

L 

\ 

R 

Tig.  246 


Fig.  245 


and  the  line  MN^  before  and  after  revolution,  are  represented  by 
tlieir  projections,  in  Fig.  246,  of  which  no  explanation  is  needed. 

316.  Applications  of  the  Preceding.  The  hyperbohc  paraboloid 
is  readily  drawn,  and  a  plane  tangent  to  it  easily  determined ;  it  is 
therefore  natural  that  this  surface  should  be  the  one  usually  em- 
ployed as  an  auxiliary,  in  constructing  tangent  plaues  to  other  and 
more  intractable  warped  surfaces,  as  suggested  in  (310):  the  two 
following  examples  illustrate  its  use  for  this  purpose. 

317.  Problem  1.  To  draw  a  plane  tangent  to  the  Cow^s  ZZorn 
at  a  given  point  of  the  surface. 


230 


DESCRIPTIVE   GEOMETRY. 


Construction.  In  Fig.  247,  X, ,  1\ ,  parallel  to  V,  are  the  cir- 
cular directrices,  whose  centres  JT  and  Y,  as  well  as  tlie  rectilinear 
directrix  ^iT,  lie  in  the  plane  JJ  parallel  to  H ;   it  is  required  to 


Fia.  24:7 


draw  a  plane  tangent  to  the  surface  at  the  point  P,  upon  the  given 
element  FD, 

This  element  and  the  rectilinear  directrix,  being  two  lines  of 
the  surface,  determine  a  plane  gQm\  tangent  thereto  at  their  inter- 
section K  Draw  in  this  plane  a  line  ^^  parallel  to  V,  and  at  D 
and  F  draw  tangents  to  the  circulai-  directrices ;  these  three  lines 


DESCRIPTIVE    GEOMETRY.  221 

are  elements  of  one  generation,  and  ED  is  an  element  of  the  other 
generation,  of  a  hyperbolic  paraboloid  liaving  V  for  one  plane 
director. 

Through  any  point  B  of  the  tangent  at  i>,  pass  a  plane  con- 
taining the  tangent  at  F\  this  plane  cuts  EK\vl  (7,  and  CTJR  is  an 
element  of  the  paraboloid. 

Through  P  pass  the  plane  //  parallel  to  T,  cutting  Cli  in  O^ 
then  PO  and  EB  determine  a  plane  tangent  at  P  to  the  auxiliary 
paraboloid  and  therefore  to  the  given  surface.  ED  pierces  H  in 
G^  and  PO  pierces  it  in  S\  also  ED  j)ierces  V  in  J[f :  consequently 
tT^  the  horizontal  trace  of  the  required  plane,  transverses  g  and  «, 
and  Tt\  the  vertical  trace,  passes  through  m' — being,  moreover, 
parallel  io  jp'o'  the  vertical  projection  oi  PO. 

318.  Problem  2.  To  draw,  a  plane  tangent  to  the  cylindroid 
at  a  given  i)oint  of  the  surface. 

Construction.  In  Fig.  248,  let  1",  Z,  be  the  circular  directrices, 
V  the  plane  directer,  and  P,  on  the  element  CD^  the  given  point. 
Draw,  at  C  and  Z>,  tangents  to  the  circles ;  these  tangents  are  the 
directrices,  and  CD  is  the  generatrix,  of  the  auxiliary  hyperbolic 
paraboloid.  The  tangents  pierce  y  m  E  and  E  respectively,  and 
EE  is  another  element  of  the  auxiliary  surface.  Pass  through  P 
a  plane  parallel  to  Z^jB'and  CE\  its  vertical  trace  is  r's'^  which  cuts 
ef  in  o\  horizontally  projected  in  o  on  AB.  Then  PO  and  CD 
determine  the  required  tangent  plane;  whose  vertical  trace  is 
l^arallel  to  c'd'  the  vertical  projection  of  CD. 

319.  The  normals  to  any  warped  surface  at  points  of  a  given 
element  thereof,  determine,  in  general,  a  hyperbolic  paraboloid 
(315),  and  belong  to  the  same  generation.  If  any  element  of  the 
other  generation  of  this  paraboloid  be  taken  as  an  axis,  the  given 
element  by  revolving  around  it  will  generate  an  hyperboloid  of 
revolution,  which  will  be  tangent  to  the  given  surface  all  along  the 
element :  of  which  the  following  exhibits  a  useful  application  «in 
mechanism. 

320.  Problems.  To  construct  two  hyperholoids  of  revolution^ 
tangent  to  each  other  along  an  element. 

Construction.  In  Fig.  249,  let  the  axis  of  one  hyperboloid  be 
vertical,  c  being  its  horizontal  and  o'c'  its  vertical  projection ;  let 


222 


DESCRIPTIVE    GEOMETRY. 


CO  be  tlie  radius  of  its  gorge  circle,  and  OP^  parallel  to  T,  its  gen- 
eratrix :  the  projections  of  this  surface  are  then  drawn  as  in  Fig. 
209.  The  gorge  radius  of  which  o'  is  the  vertical  and  co  is  the 
horizontal  projection,  is  evidently  normal  to  the  surface ;  the  nor- 
mal at  P  must  lie  in  a  plane  perpendicular  to  OP^  therefore  c'p' 
perpendicular  to  o';p'  is  its  vertical  and  cj)  is  its  horizontal  projec- 


tidh :  these  normals  are  elements  of  one  generation,  and  the  verti- 
cal axis  and  the  generatrix  OP  are  elements  of  the  other  generation 
of  the  normal  hyperbolic  paraboloid, — of  which  one  plane  director 
is  V,  and  the  other  is  perpendicular  to  OP.  Any  plane  as  JJ 
parallel  to  Y  is  seen  in  the  horizontal  projection  to  cut  co  produced, 
in  6,  and  cp  produced,  in  ^;   e  is  vertically  projected  in  o' ^  and  d 


DESCRIPTIVE    GEOMETRY. 


223 


in  d\  on  the  prolongation  of  c'])\  therefore  ed  is  tlie  horizontal,  and 
o'd'  the  vertical,  projection  of  another  element  of  the  normal  parab- 
oloid :  which  may  be  taken  as  the  axis  of  the  second  hyperboloid. 

321.  Make  a  supplementary  projection  on  a  plane  perpendicular 
to  this  axis,  looking  in  the  direction  of  the  arrow  w.     In  this  view^ 


eo  is  seen  in  its  true  length  as  ^^<9, ,  and  the  common  element  OP 
as  (9i^j  tangent  at  o,  to  the  circle  of  the  gorge,  which  in  the  verti- 
cal elevation  appears  2,^  fk!  perpendicular  to  o'd' .  The  radius  of 
the  upper  base,  passing  "  through  P  and  also  perpendicular  to  the 


224  DESCRIPTIYE   GEOMETRY. 

inclined  axis,  is  e^p^ ,  to  which  accordingly  g'V  in  the  vertical  pro- 
jection is  made  eqnal.  Any  point  TJ  on  OP^  is  projected  to  %i^  on 
o^])^  ,  and  s'li't'  perpendicular  to  o'd!  is  eqnal  to  e(ii^ ;  and  in  like 
manner  any  desired  number  of  points  on  the  required  hyperbola 
may  be  found. 

In  the  foreshortened  horizontal  projection  of  the  inclined  hyper- 
boloid,  both  the  gorge  circle  and  the  upper  base  appear  as  ellipses ; 
as  will  also  any  intermediate  transverse  sections^.  A  portion  of  one 
such  section  through  B^  is  projected  at  y ;  the  visible  contour  zx  is 
the  enyelope  of  all  these  ellipses,  and  not,  as  sometimes  supposed,  a 
curve  through  the  extremities  of  their  major  axes :  in  fact  it  passes 
through  the  extremity  of  only  one  of  them,  viz. ,  that  of  the  gorge 
circle  at  2,  at  which  point  the  contour  line  has  a  tangent  perpen- 
dicular to  ez  the  gorge  radius. 

INTERSECTIONS    OF    WARPED    SURFACES. 

322.   The  intersection  of  any  warped  surface  with  a  plane  may 

be  determined  by  finding  the  points  in  which  its  elements  pierce 
the  plane.  Many  such  intersections  have  already  been  illustrated ; 
and  in  any  case  the  problem  is  simple  in  principle,  though  the 
necessary  repetition  of  the  same  process  may  render  it  tedious  in 
execution. 

The  intersection  with  any  other  surface  may  be  determined  by 
the  general  method,  of  passing  a  series  of  auxiliary  planes  cutting 
both  surfaces,  and  joining  the  points  in  which  the  lines  cut  from 
each  surface  intersect  each  other.  Just  what  system  of  auxiliary 
planes  will  be  most  convenient,  must  in  the  nature  of  things  dej^end 
largely  upon  the  peculiarities  of  any  given  case,  and  be  decided  by 
the  judgment  of  the  operator.  Attention  to  the  following  points 
may,  however,  sometimes  be  of  service ; — 

1 .  If  one  of  the  given  surfaces  he  a  cylinder^  Planes  may  be 
passed  through  the  elements  of  the  warped  surface,  parallel  to 
those  of  the  cylinder. 

2.  If  one  of  the  s  117 faces  he  a  cone;  Planes  may  be  passed 
through  the  elements  of  the  warped  surface  and  the  vertex  of  the 
cone. 

3.  If  hoth  surfaces  are  war][)ed^  hut  have  a  common  plane  di- 


DESCRIPTIVE    GEOMETRY. 


325 


recterj  A  system  of  planes  parallel  to  this  plane  directer  may  be 
used. 

In  either  of  these  .cases,  the  auxiliary  planes  will  cut  right  lines 
from  both  surfaces ;  but  it  does  not  necessarily  follow  that  these 
wdll  give  the  most  satisfactory  determinations,  since  they  may  in- 
tersect each  other  very  acutely. 

323.  The  intersection  of  a  helicoid  with  a  surface  of  revolution 
having  the  same  axis,  is  of  special  interest  as  being  frequently  met 
with  in  the  construction  of  screw-propellers ;  a  few  illustrations  of 
it  are  therefore  appended.  That  particular  form  of  the  helicoid 
only  is  here  considered  in  which  the  generatrix  cuts  the  axis ;  bo- 


rn' 

\d"   y"    h" 


cause  in  practice  it  is  used,  if  not  exclusively,  at  least  more  exten* 
sively  than  any  other. 

324.  Example  1.  Intersection  of  a  helicoid  with  a  right  cir- 
cular cone  having  the  same  axis. 

Construction.  In  Fig.  250  are  given,  on  the  left  an  end  view, 
on  the  right  a  side  view^,  of  a  portion  of  a  right  helicoid ;  ad^  ad\ 
being  the  helical  directrix.  In  the  end  view,  ca^  cb^  cd^  etc.,  equi- 
distant radii,  represent  elements;  if  these  be  revolved  into  the  ver- 
tical plane  ca^  they  will  in  the  side  view  appear  as  the  equidistant 
lines  G'a\  r'h'\  w'd'\  etc.,  perpendicular  to  the  axis;  these  pierce 
the  cone  m'o'n'  at  the  points  h\  e'\  d'\  etc.      Set  off  on  the  radii 


226 


DESCRIPTIVE    GEOMETRY. 


in  tlie  end  view  the  true  distances  of  these  points  from  the  axis,  as 

cJi  =:  c'h\  ce  =  r'e'\  ct  =  s't" ^  etc.  Project  the  points  thus 
located,  back  to  the  elements  in  the  side  view,  as  d  Xo  d\  t  to  t\ 
e  to  e\  etc.  ;  the  curves  deli^  d'e'K ^  thus  determined,  are  the  re- 
quired projections  of  the  intersection. 


Tig.  252 


In  the  case  of  the  oblique  helicoid,  Fig.  251,  the  elements  when 
revolved  into  the  vertical  plane  ca^  appear  in  tlie  side  view  as  equi- 
distant parallels  inclined  to  the  axis. 

Tlie  construction  is  the  same  as  before,  with  the  exception  that 
the  points  d\  t\  e' ^  etc.,  are  located,  not  upon  tlie  elements,  but 
upon  perpendiculars  to  the  axis  from  the  points  d'\  t'\  e'\  etc.  ; 
because  each  point  must  revolve  in  a  plane  pb/pendicular  to  the 
axis. 


DESCRIPTIVE   GEOMETRY. 


227 


Fig.  252  represents  tlie  intersection  of  the  cone  with  a  hehcoid 
of  varying  pitcli.  In  this  case  the  revolved  elements  appear  in  the 
side  view  as  lines  of  different  inclinations,  dividing  into  the  same 
number  of  equal  parts  the  distances  a'd'\  c'w'\  which  distances 
are  the  same  fractions  of  the  ]^itches  at  tlie  outer  circnrnforence  and 
the  axis  respectively,  that  tlie  arc  ad  is  of  the  whole  circumfer- 
ence :  otherwise  the  construction  is  the  same  as  in  Fig.  251. 

325.  Note.  The  arcs  hg^  ef^  etc.,  in  the  end  views,  represent 
concentric  cylinders,  which  cut  all  these  helicoids  in  true  helices — 
two  of  them  shown  in  the  side  views  as  k'g\  e'f.     The  outlines 


ym.  253 


y'g'  ^  e"f'^  of  these  cylinders  must  pass  through  the  points  Tc"^  e'\ 
i" ^  etc.,  in  all  three  side  views;  in  the  first  two  these  points  are 
equidistant,  therefore  in  the  end  views  the  points  A,  g^  y,  are  also 
equidistant,  and  the  curve  deli  in  Figs.  250  and  251  is  an  Archi- 
medean spiral :  but  in  Fig.  252  this  is  not  so,  for  the  distances  hg^ 
gf^  etc.,  are  not  equal,  but  continually  increase  as  the  points  A,  ^,/*, 
recede  from  the  centre. 

This  problem  is  encountered  in  determining  the  form  of  the 
trailing  edge  of  a  propeller-blade  having  what  is  technically  called 
an  ' '  overhang ' ' ;   when  it  will  often  be  found  more  satisfactory  to 


228  DESCRIPTIVE   GEOMETRY. 

ascertain  the  distances  ck^  ce^  etc. ,  by  calculations  based  upon  tlie 
law  of  tlie  spiral,  since  accuracy  in  tlie  end  view  is  most  essential. 

326.  Example  2.  Intersection  of  a  helicoid  with  an  annular 
torus  having  the  same  axis. 

Explanatory.  A  propeller-blade  fashioned  as  above  would  re- 
volve within  a  surface  of  revolution  whose  meridian  section  is 
o'd"a'  in  either  of  the  three  preceding  figures:  it  would  also  have 
an  objectionable  sharp  corner  at  d^  d' ,  The  imaginary  "box" 
within  which  the  blade  revolves  is  therefore  sometinies  '  rounded 
off  at  the  angle,'  as  by  the  arc  x'\(!'n"  in  Fig.  253,  This  arc  is  a 
part  of  the  circumference  of  a  circle  whose  centre  is  z ;  and  the 
complete  circumference  is  the  meridian  section  of  an  annular  torus 
having. the  same  axis  as  the  helicoid. 

Construction.  The  w^arped  surface  in  this  figure  is  a  right  lieli- 
coid,  and  the  mode  of  operation  is  substantially  the  same  as  in  Fig. 
250.  Thus,  dividing  the  arc  ad  and  the  distance  a'd"  in  the  same 
proportion,  the  elements  are  represented  by  radii  in  the  end  view 
and  ])y  perpendiculars  to  the  axis  in  tlie  side  view,  drawn  through 
the  points  of  division;  and  the  curves  deh.,  d'eh\  are  found  as  in 
(324).  The  section  of  the  torus  is  tangent  at  ti"  to  the  element 
s'y" ;  set  off  on  cy,  the  distance  cu  =  s'u'\  and  project  u  back  to 
s'y"  at  It' ;  this  determines  a  limiting  point  of  the  curve  in  each 
view.  The  circular  section  is  tangent  to  o'7ri  at  v'\  and  an  ele- 
ment through  this  point  cuts  the  circumference  also  at  t" :  the 
distances  of  these  points  from  the  axis  being  set  ofl:  on  the  corre- 
sponding radius,  the  two  points  t  and  v  are  determined — of  which 
the  latter  is  the  point  of  tangency  betw^een  the  spiral  aeh  and  the 
new  intersection  vtx.  Ry  repeating  this  process  as  many  points  as 
are  deemed  requisite  may  be  found,  and  the  entire  curve  of  pene- 
tration constructed :  only  that  portion  is  here  shown  wdiich  would 
form  part  of  the  contour  of  the  blade  of  a  propeller. 

327.  Example  3.  Intersection  of  a  helicoid  v&ilh  a  cylinder 
whose  elements  are  parallel  to  the  axis. 

Explanatory.  The  form  of  a  propeller  blade  is  sometimes  fixed 
by  the  condition  that  it  shall  present  a  given  outline  in  the  end 
view :  the  drawing  of  the  other  views  then  involves  the  problem 
above  mentioned. 


DESCRIPTIVE    GEOMETRY. 


229 


Construction.  In  Fig.  254,  let  the  warped  surface  again  be  a 
right  heHcoid,  and  let  mon  be  the  base  of  the  cylindrical  surface. 

The  operation  is  simply  the  converse  of  the  preceding ;  the 
points  in  which  the  elements  of  the  helicoid  pierce  the  cylinder  are 
seen  directly  in  the  end  view,  and  are  projected  to  the  correspond- 
ing elements  as  seen  in  the  side  view, — as  e  to  e'  on  w'd" ^  o  to  o'  on 
s'y" ,  etc.     Then  in  order  to  find  the  outline  of  the  surface  within 


FiGf.  254 


which  the  blade  revolves,  set  up  w'e"  =  ce^  s'o"  =  co,  and  so  or>, 
thus  determining  the  required  curve  e"o"x" . 

328.  These  last  problems  have  been  illustrated  only  in  connec- 
tion with  the  riglit  helicoid,  merely  for  tlie  sake  of  simplicity. 
But  by  attention  to  the  explanations  given  in  (324),  there  will  be 
no  difficulty  in  dealing  in  a  similar  manner  with  the  others ;  and 
indeed  substantially  the  same  methods  might  be  applied  if  the  gen- 
eratrices of  the  helicoids  were  curved,  as  they  sometimes  are :  in 
whicli  case,  however,  the  surfaces  are  no  longer  warped,  but  are  of 
double  curvature. 


230 


DESCmPTIVE   GEOMETET. 


CHAPTEK  YII. 

ISOMETRICAL      DeAWING,     CavALIER      PROJECTION,     AND      PsEtJDO- 

Perspective. 


ISOMETRY. 


329.  In  Fig.  255,  (7  is  a  top  view  of  a  cube  so  placed  that  in 
the  front  view  A  tlie  diagonals  cc/,  ah,  of  its  upper  face  are  respec- 
tively parallel  and  perpendicular  to  the  paper.     The  cube  is  shown 


as  cut  by  a  plane  pp,  perpendicular  to  the  paper  in  view  A ;  the 
gection  thus  made,  as  seen  in  the  perspective  sketch  F,  is  bounded 
"by  the  three  face  diagonals  ah,  ad,  hd :  it.  is,  then,  an  equilateral 
triangle,  to  the  plane  of  which  the  three  equal  edges  ca,  ch,  cd  are 


DESCRIPTIVE    GEOMETRY.  231 

e' iially  inclined.  And  as  seen  in  view  ^,  this  plane  is  perpen- 
dicular to  the  body  diagonal  ch  of  the  cube,  which  pierces  it  at  o. 

In  the  view  i>,  which  is  an  orthographic  projection  upon  the 
plane  ^j?,  the  three  face  diagonals  are  seen  in  their  true  lengths, 
forming  the  equilateral  triangle  a'Vd' .  Since  the  three  edges 
which  meet  at  g  are  equally  inclined  to  the  plane,  they  will  be 
equally  foreshortened :  therefore  c'  is  the  centre  of  the  triangle, 
a'G\  h'c'^  d'c'  are  equal  to  each  other,  and  the  three  angles  at  c'  are 
each  equal  to  120°. 

Every  other  edge  of  the  cube  being  equal  and  parallel  to  one  of 
these  three,  each  visible  one  will  appear  equal  and  parallel  to  one 
of  those  already  drawn ;  thus  the  apparent  contour  of  the  entire 
cube  will  be  a  regular  hexagon,  the  representation  of  each  face  be- 
ing a  rhombus. 

Because  the  edges  of  the  cube  are  thus  foreshortened  in  the 
same  proportion,  so  that  they  and  all  parallels  to  them  may  be 
measured  l)y  the  same  scale,  such  a  view  as  D  is  called  an  Isometric 
Projection ;  a'c' ^  h' g' ^  d'c\  are  called  the  isometric  axes  ;  the  planes 
which  they  determine,  and  all  planes  parallel  to  them,  are  called 
isometric  planes  /  and  all  lines  parallel  to  the  axes  are  called  iso- 
metric  lines. 

330.  Drawings  made  in  this  manner  possess  the  advantage  of 
conveying,  in  one  view,  ideas  of  the  three  dimensions,  as  do  those 
made  in  perspective ;  and  in  many  cases  they  exhibit  the  peculiarities 
of  structure  more  clearly  than  ordinary  plans,  sections,  and  eleva- 
tions. They  are  readily  understood  by  those  who  are  not  familiar 
with  common  projections ;  and  in  making  sketches  this  system  is 
very  useful. 

Obviously,  however,  the  advantages  of  isometry  are  more  pro- 
nounced Avhen  the  objects  to  be  represented  are  bounded  by  right 
lines,  of  which  the  principal  ones  are  parallel  and  perpendicular  to 
each  other.  It  is  not  well  adapted  for  the  general  drawing  of 
machinery,  since  it  involves  an  unpleasant  distortion,  and  also  be- 
cause in  most  cases  the  circles  are  projected  as  ellipses. 

331.  Distinction  hetween  isonietrical  projection  and  isometri- 
col  drawing.  In  Fig.  255  the  actual  length  of  the  edge  of  the  cube 
is  cd\  its  apparent  length  in  view  D  is  c'd' ^  equal  to  od  in  view  A, 


23^  DESCRIPTIVE    GEOMETRY. 

Suppose  cd  to  be  one  unit  in  length — an  incli  for  example :  tlien 
by  taking  od  as  a  unit  it  is  possible  to  construct  an  isometriG  scale^ 
by  wliicli  all  tlie  isometric  lines  in  D  might  have  been  set  off ;  and 
such  a  scale  could  be  used  in  constructing  any  isometrical  projection. 

This  is  a  matter  of  purely  abstract,  theoretical  interest,  and  not 
of  any  practical  use  whatever.  Since  tlie  isometric  lines  are  all 
equally  f-oreshortened,  there  is  no  reason  why  they  should  be  repre- 
sented ac  foreshortened  at  all.  Consequently  an  Isometric  Drawing 
of  the  given  cube  is  made  as  shown  at  E^  each  edge  being  drawn  of 
its  true  length.  This  is  the  method  always  adoj^ted  in  practice, 
the  scales  in  common  use  being  alone  employed.  The  man  who 
should  construct  a  true  projeetio7i,  and  send  it  to  the  workman  to 
be  measured,  by  an  isometric  scale,  would  simply  make  a  record  of 
his  own  stupidity ;  he  who  sliould  teach  others  to  do  so,  would 
commit  a  blunder  of  much  more  serious  importance.  For,  to  use 
the  words  of  another,  ''  the  value  of  isometry  as  a  practical  art  lies 
in  the  applicability  of  common  and  known  scales  to  the  isometric 
lines.  "-^ 

332.  We  proceed,  then,  just  as  in  making  ordinary  working 
drawings,  setting  off  the  dimensions  on  those  lines  either  ' '  full 
size,"  or  with  the  3-inch  scale,  the  li-inch  scale,  etc.,  as  the  case 
may  require.  Naturally,  lines  which  are  vertical  are  so  repre- 
sented ;  the  other  isometric  lines  are  then  drawn  with  great  facility 
by  the  aid  of  the  T-square  and  the  triangle  of  60°  and  30°. 

Figs.  256-261  are  simple  exercises,  composed  wholly  of  iso- 
metric lines,  the  construction  being  so  obvious  that  no  detailed 
explanation  is  required :  the  method  of  locating  the  foot  of  the 
cross  in  Fig.  257,  and  the  mortise  and  the  tenon  in  Fig.  261,  by 
measui'ing  along  the  lines  ea,  ch,  or  parallels  to  them,  is  sufficiently 
shown  by  the  dotted  lines. 

It  is  to  be  distinctly  understood  that  these  figures  are  illustra- 
tions merely :  the  student  is  not  to  coj>y  them,  but  to  construct 
them  or  others  of  similar  character,  with  such  variations  of  dimen- 
sions, arrangement,  or  design  as  may  be  suggested  by  his  ingenuity, 
which  should  be  given  full  play. 

*  W.  E.  Wortben. 


DESCRIPTIVE   GEOMETRY. 


!33 


333.    Shadow  Lines.     In  making  mechanical  drawings,  on  any 
system  of  projection,  those  lines  which,  being  the  intersections  of 


Fig.  257 


snrfaces  which  are  illuminated  w^ith  others  which  are  not,  intercept 
the  light  and  thus  cast  shadows,  are  usually  emphasized  by  making 


Fm.  258 


them  heavier  than  the  other  outlines.     By  this  means  an  effect  of 
relief  is  produced ;  the  drawing  is  more  easily  read,  and  its  ajDpear- 


J34 


DESCRIPTIVE   GEOMETRY. 


ance  greatly  improved :  and  tlie  lines  tliiis  emphasized  are  called 
shadow  lines. 

In  orthographic  drawings,  the  direction  of  the  light  is  by  com- 
mon consent  assumed  as  follows :  Suppose  the  observer  to  be 
standing  in  a  cubical  room,  facing  one  of  the  walls  as  the  vertical 
plane ;  then  the  light  comes  from  behind,  the  rays  going  downward 
to  the  right,  in  the  direction  of  the  body  diagonal  of  the  cube,  each 
projection  making  an  angle  of  45°  with  the  ground  line. 

And  in  isometric  drawings  reference  is  also  made  to  the  cube 
as  a  standard.  Thus  in  E^  Fig.  255,  the  light  is  supposed  to  have 
the  direction  of  the  body  diagonal  af^  so  that  the  faces  eo^  cg^  are 


Fig.  261 


illuminated,  and  shadows  are  cast  by  the  edges  ed^  dc^  cb^  and  Ig, 
In  drawing,  the  first  of  these  lines  should  be  made  the  heaviest, 
the  last  one  the  lightest,  and  the  other  two  of  equal  and  medium 
thickness. 

334.  Isometrical  Drawing  of  the  Circle — In  the  ellipse  repre- 
senting the  circle  inscribed  in  the  face  of  the  cube.  Fig.  262,  the 
axes  coincide  with  the  diagonals,  and  are  at  once  determined  by 
representing  the  parallels  through  Z,  m,  n^  6>,  in  the  elevation  shown 
at  the  left.  Describe  a  semicircle  upon  cd  as  a  diameter,  divide  it 
into  four  equal  parts  by  the  points  1,  2,  3,  draw  3  3',  1  1'  perjDen- 
dicular  to  cd^  and  through  3'  and  V  draw  parallels  to   hc]  tliese 


DESCRIPTIVE    GEOMETRY. 


235 


will  cut  the  diagonals  at  m,  <?,  and  n^  I,  thus  limiting  the  major  and 
minor  axes.  As  a  check,  note  that  hn  and  on  should  be  parallel 
to  cd. 

The  sides  of  the  rhombus  being  equal,  tins  construction  may  be 
made  upon  either  one  at  pleasure.  And,  since  all  the  faces  of  the 
cube  are  exactly  alike,  it  follows  that  all  circles  lying  in  isometric 
planes  are  represented  by  similar  ellipses. 

By  drawing  tangents  at  the  points  ?,  in^  n,  o  in  the  elevation  the 
circle  is  circumscribed  by  a  regular  octagon,  the  isometric  represen- 
tation of  which  is  therefore  made  by  drawing  at  the  corresponding 


Tig.  262 


points  Z',  m',  n\  o\  in  the  upper  face  of  the  cube,  perpendiculars  to 
the  diagonals,  terminating  in  the  sides  of  the  rhombus. 

335.  Graduation  of  the  Isometric  Circle.  First  Method.  At 
the  middle  point  d  of  gh^  in  Fig.  263,  erect  a  perpendicular  de 
equal  to  dh^  and  draw  eh.,  eg ;  about  ^  as  a  centre  describe  with  any 
radius  the  quadrant  r^,  divide  it  as  desired  by  the  points  1,  2,  3, 
etc.,  through  which  draw  radii  and  produce  them  to  cut  gh. 
From  these  intersections  with  gh  draw  lines  to  ^,  the  centre  of  the 
ellipse :  these  will  cut  its  circumference  in  the  required  points  of 
division,  1,,  2, ,  etc. 

Second  Method.  Describe  a  semicircle  upon  the  major  axis  ttio 
as  a  diameter,  divide  it  in  the  desired  manner  by  the  points  1',  2', 


236 


DESCRIPTIVE   fiEOMETRY. 


etc. ,  tlirongli  wliicli  draw  perpendiculars  to  mo^  cutting  tlie  cir- 
cumference of  the  ellipse  in  1',  2',  etc.  :  these  will  be  the  points 
required. 


Fig.  263 


An  application  of  the  above  is  shown  in  the  drawing  of  the 
bolt,  nut,  and  washer,  Fig.  264.  About  ^,  the  centre  of  the  outer 
ellipse,  describe  an  arc  with  radius  j96>  =  semi-major  axis,  set  ofi  the^ 


DESCRIPTIVE   GEOMETRY. 


237 


arc  oa  =  60°,  erect  tlie  vertical  ab^  and  draw  h])  cutting  the  inner 
ellipse  (circnmscri})ing  the  base  of  the  nut)  in  c. 

336.  To  draw  Angles  to  the  Sides  of  the  Isometrical  Cube  (Fig. 
265).  Draw  a  square  cg^  whose  side  is  equal  to  the  edge  of  the 
cube ;  about  one  of  its  angles,  say  <?,  as  a  centre,  describe  the  quad- 
rant db^  graduate  it,  and  produce  the  radii  through  the  points  of 
division  to  cut  the  sides  of  the  square.  The  scale  of  tangents  thus 
formed  may,  bj  cutting  out  the  square,  be  applied  to  any  side  of 
the  isometrical  cube,  thus  determining  the  direction  of  a  line  in  its 
face  which  shall  represent  a  line  making  any  required  angle  with 
its  edge.     For  example,  make  a'e'=ae^  and  h'f'=  hf\  then  a'o'e'^ 


Pig.  265 


y 


Vcf  are  the  isometrical  representations  of  the  angles  ace^  hcf. 
The  same  angles  are  represented  on  the  left-hand  vertical  face  of 
the  cube  by  making  d'e"=  ae^  G^f"=  ^?  and  drawing  a^e^'^  (^'f* 

An  application  of  this  is  found  in  making  the  isometrical  draw- 
ing of  the  piece  shown  in  plan  and  elevation  at  6^  and  A^  Fig.  266, 
in  which  the  angles  a?,  y  are  assigned :  also  in  C  the  distance  ad 
is  given,  dke  is  perpendicular  to  ad^  and  ef  parallel  to  ok :  the 
thickness  is  uniform,  and  equal  to  ah  in  view  A.  The  isometric 
drawing  is  lettered  to  correspond,  and  should  require  no  further 
explanation. 

337.  Another  method  of  dealing  with  lines  which,  though 
lying  in  isometri  3  planes,  are  not  parallel  to  either  of  the  isometric 
axed,  is  by  means  of  ^'  offsets."  Thus  in  Fig.  267  the  slope  of  the 
diagonal  brace  is  determined  by  measuring  the  distances  cd,  d^, 


238 


DESCRIPTIVE    GEOMETRY. 


along  tlie  isometric  line  ca^  and  setting  np  the  vertical  ef^  of  the 
values  ascertained  from  tlie  elevation  shown  on  a  reduced  scale. 
This  really  amounts  to  the  same  thing,  the  angle  being  constructed 
bj  laying  off  the  base  and  altitude  of  a  triangle  of  which  the  re- 
quired line  is  tlie  hypothenuse, — which  is  in  perhaps  the  majority 
of  cases  the  most  convenient  means.  Another  illustration  is  given 
in  the  drawing  of  the  box,  Fig.  268 ;  the  outline  of  the  end  of  the 
partially  opened  lid  being  set  out  by  means  of  the  vertical  meas- 
urements <?«•,  co^  cb,  ch^  and  the  offsets  ae^  ol^  hcl^  hg^  taken  directly 


Fig.  268 


from  the  transverse  section  shown  at  the  left. 

338.  This  principle  may  be  extended,  and  is  applied  to  the 
determination  of  lines  which  do  not  lie  in  isometric  planes ;  as  illus- 
trated in  Fig.  269,  representing  the  roof  of  a  cottage,  cf  the  fonn 
and  proportions  shown  in  plan  and  elevations  on  a  smaller  scale  at 
the  right.  The  sloping  lines  of  the  roof  at  the  nearer  end  are 
found  by  setting'  up  the  heights  ah^  ad  on  the  vertical  through  a,, 
and  drawing  the  isometric  lines  JA,  df:  then  the  points  ?',  Jc  are  the 
intersections  of  hf^fh  with  the  isometrical  line  through  c.  A  simi- 
Jar  construction  may  be  made  at  the  farther  end,  thus  fixing  the 
line  of  the  ridge  ^',  on  which  the  point  g  is  located  by  setting  olf 


DESCRIPTIVE    GEOMETRY. 


239 


fg^  its  distance  from  the  plane  de :  we  can  then  draw  gi^  gh^  which 
do  not  lie  in  any  isometric  plane. 

The  same  process  is  applied  in  drawing  the  wing  roof,  the 
heights  n^  v,  r^  being  set  up  on  the  vertical  through  the  nearer 
corner  ??^,  and  the  distance  oq  measured  from  the  plane  mo.  The 
ridge  line  w^ill  pierce  the  main  roof  at  a  point  t^,  which  may  be 
thus  located :  Set  up  at  =  mr,  draw  the  isometric  line  ts  cutting 
^"in  6',  and  through  s  draw  a  parallel  toff:  this  will  cut  the  ridga 
line  of  the  wing  in  the  required  point. 

It  will  be   observed  that  the  lines  fh^  gl^^  and  ff\  differ  very 


Pia.  269 


JP  ^ -. 


m 


little  in  direction,  and  qz.,  ou^  differ  still  less.  I'his  simply  shows 
that  the  farther  side  of  the  main  roof  is  very  nearly,  and  that  of 
the  w^ing  roof  almost  exactly,  perpendicular  to  the  plane  upon 
which  the  isometric  drawing  is  made ;  and  it  will  be  perceived  that 
in  such  cases  this  not  a  peculiarly  eligible  mode  of  representation, 
— as  indeed  it  is  not  for  architectural  subjects  of  any  description. 

339.  Thus  far  one  of  the  isometric  axes  has  been  made  vertical. 
But  inasmuch  as  it  is  the  relative  direction  of  the  lines  among 
themselves  which  determines  whether  a  drawing  is  an  isometric  one 
or  not,  there  is  no  necessity  that  any  of  them  should  be  vertical. 
In  Fig.  270,  for  example,  the  principal  lines  are  horizontal ;  but 
the  drawings  of  the  die  and  its  matrix,  and  of  the  timber  with  its- 


340  DESCRIPTIVE   GEOMETRY. 

mortises  and  its  tenon,  are  at  once  recognized  as  isometric,  and 
are  just  as  easily  understood  as  if  tliej  stood  upiight. 

For  convenience  in  constructing  the  drawings  by  means  of  the 
T-square  and  triangles,  it  is  preferable  in  most  cases,  of  course,  that 
one  of  the  isometric  axes  should  be  either  vertical  or  horizontal,  but 
should  there  be  any  reason  for  selecting  other  positions,  there  is 
nothing  in  the  principle  of  isometry  to  prevent  their  adoption. 

340.  It  will  be  noted  that  the  correspondence  of  the  die  to  the 
matrix  in  Fig.  270  is  made  much  more  obvious  than  it  otherwise 
would  be,  by  exhibiting  the  opposite  ends  of  the  tw^o  pieces.     By 


V 
^^i     VA\     'CIA 


Fig.  270 

merely  turning  the  page  around,  it  will  be  apparent  that  this  could 
have  been  done  equally  w^ell  if  the  two  pieces  had  been  drawn  in  a 
vertical  position. 

For  this  reason  isometry  affords  a  means  of  illustrating  in  a  very 
clear  and  striking  manner  many  subjects  in  which  views  of  the 
lower  surfaces  are  desirable  :  a  good  example  is  shown  in  the  draw- 
ing of  the  small  shelf  with  its  supporting  bracket.  Fig.  271. 

In  making  such  a  drawing,  as  will  readily  be  seen,  the  process 
is  equivalent  to  constructing  the  projection  of  the  cube,  Fig.  255. 
upon  the  plane  ^j!?,  as  seen  from  the  lower  left-hand  side,  and  look 
ing  in  the  direction  opposite  to  that  indicated  by  the  arrow. 

341.  In  Fig.  272  is  shown  an  isometric. drawing  of  the  ratchet- 
wheel  represented  in  the  full-size  views  at  the  right.     The  backs  of 


DESCRIPTIVE   GEOMETRY. 


241 


the  teetli  not  only  terminate  in,  but  are  tangent  to,  tlie  interior 
circle ;  and  a  test  of  the  accuracy  of  the  isometric  drawing  is  found 
in  the  tangency  of  the  edges  to  the  ellipse  representing  that  circle. 
And  this  embodies  a  principle  capable  of  many  other  applications 
— as,  for  instance,  in  laying  out  a  wheel  with  radial  tapering  arms  : 


Fig.  271 


the  side  outlines  of  each  arm  are  tangent  to  a  circle,  which  being 
drawn  in  the  isometric  construction,  it  is  seen  that  the  breadths  of 
the  a'rms  at  the  outer  ends  only  need  be  set  out,  thus  fixing  points 
through  which  tangents  are  to  be  drawn  to  the  ellipse. 

342.  It  seems  needless  to  multiply  examples,  as  it  is  believed 
that  by  the  aid  of  the  preceding  any  isometrical  drawing  likely  to 
be  required  in  practice  may  be  constructed. 


In  drawings  of  machinery,  the  circles  of  wheels,  bearings,  ends 
of  shafts,  and  the  like,  will  usually  lie  in  isometric  planes.  Should 
occasion  arise  to  represent  one  which  does  not,  circumscribe  it  by 
a  square :  the  projection  of  this  will  be  a  parallelogram,  within 
-)Yhich  the  ellipse  may  be  drawn  by  any  convenient  method.      So, 


242 


DESCRIPTIVE    GEOMETRY. 


too^  if  it  should  be  necessary  to  represent  tlie  section  of  a  cylinder 
or  a  cone  by  an  oblique  plane,  the  solid  may  be  conceived  as  sur- 
rounded by  a  square  pyramid  or  prism,  whose  section  by  the  given 
plane,  as  well  as  tlie  isometric  drawing  of  it,  will  be  a  parallelo- 
gram circumscribing  the  required  ellipse. 


343.  In  conclusion,  it  may  be  pointed  out  that  many  lines  not 
usually  classed  as  isometric  are  strictly  so  in  fact.  This  distinctive 
term  is  technically  restricted  to  lines  parallel  to  the  isometric  axes, 
which  again  are  so  called  because  they  are  equally  foreshortened, 
and  this  is  the  result  of  their  equal  inclination  to  the  plane  upon 


DESCRIPTIVE   GEOMETRY.  ?43 

wliicli  tliey  are  projected.  Now,  in  Fig.  273  the  cube  is  cut  by 
the  plane  pp  as  in  Fig.  255,  and  in  the  projection  D  we  at  once 
recognize'  c  a\  e'h\  c'd'  as  the  isometric  axes.  If  cd  revolve  around 
€0  as  an  axis  it  will  generate  a  cone  dch^  all  of  whose  elements  make 
the  same  angle  with  the  plane  j:^^;  so  that  any  one  of  them,  as  cr 
(seen  in  D  as  c'r')^  would  be  foreshortened  in  the  same  proportion 
as  any  otlier  one.  But  in  the  isometric  projection  this  fact  w^ould 
not  be  indicated  by  merely  drawing  c'r' :  it  is  necessary  to  locate 
the  point  r'  by  means  of  offsets — g'7i  giving  its  distance  from  the 
plane  a'c'd' ^  c'l  its  distance  from  the  plane  a'o'h'^  and  n^n  its  dis- 
tance from  the  plane  h' c'd' .  This  being  done,  s'  is  at  once  s.een  to 
be  the  foot  of  the  perpendicular  r's'  from  the  point  in  question  to 
the  plane  last  mentioned. 

Again,  the  isometric  projection  of  any  frustum  of  a  cone,  xyuv, 
whose  bases  are  parallel  to  pp,  would  appear  simply  as  two  con- 
centric circles,  and  without  some  auxiliary  view  that  projection 
would  convey  no  definite  information  about  the  cone,  which  might 
be  of  any  altitude  or  have  either  base  uppermost. 

Since  the  whole  value  of  isometry,  in  practice,  lies  in  the  power 
of  imparting  in  one  view  definite  ideas  of  the  three  dimensions,  the 
above  hints  may  serve  a  purpose  as  indicating  possible  relations  of 
parts  for  the  representation  of  which  this  method  of  drawing  is  not 
suitable. 

CAVALIER    PROJECTION. 

344.  In  Fig.  274,  let  JlfiTbe  a  vertical  glass  plate  represent- 
ing the  vertical  plane ;  let  c  be  a  point  in  this  plane,  and  ca  a  line 
perpendicular  to  it.  Let  ar  be  a  visual  ray,  making  an  angle  of 
45°  with  the  plane  JO^,  and  piercing  it  at  j9 :  then  cp  is  the  repre- 
sentation of  ca  upon  the  picture  plane,  and  it  is  equal  to  ca^  be- 
cause the  angles  cpa^  cap  are  each  equal  to  45°. 

Suppose  the  eye  to  be  at  an  infinite  distance  in  the  direction  ar : 
then  all  the  visual  rays  will  be  j)arallel,  and  all  lines  perpendicular 
to  MW  will  be  represented  upon  that  plane  by  lines  of  their  actual 
length,  and  parallel  to  cp. 

The  fact  tliat  the  projection  is  of  the  same  length  as  the  per- 
pendicular line   ca   depends   upon  the  condition  that  the  picture 


244 


DESCRIPTIVE   GEOMETRY. 


plane  cuts  the  projecting  lines  at  an  angle  of  45°.  But  the  direc- 
tion of  tlie  projection  depends  upon  tliat  of  the  visual  ray.  Thus 
if  the  eye  be  still  at  an  infinite  distance,  but  in  the  direction  at^  the 
projection  will  have  tJie  direction  co^  but  its  length  will  remain 
equal  to  ca. 


Thus  the  direction  of  the  projecting  lines  may  be  parallel  to  any 
element  of  the  cone  whose  axis  is  ca^  the  angle  at  the  vertex  a  be- 
ing 90°,  since  all  these  elements  make  angles  of  45°  witli  the  pic- 
ture plane  MN. 


Fig.  275 


345.  ]^ow  any  line  which  lies  in  the  picture  plane  is  its  own 
projection.  ^  In  representing  a  cube,  therefore,  as  in  Fig.  275,  we 
may  assume  its  nearer  face  to  lie  in  that  plane,  and  it  will  thus  ap- 
pear of  its  true  form  and  size,  that  is,  a  square,  as  shown.  From 
the  preceding  it  follows  at  once  that  the  edges  which  are  perpen- 
dicular to  MN  may  be  represented  by  parallel  lines  of  their  true 


DESCRIPTIVE    GEOMETRY. 


245 


lengtli,  but  having  any  direction  at  pleasure ;  which  enables  us  to 
show,  in  addition  to  the  front  face,  either  the  right  face  or  the  left, 
the  upper  or  the  lower,  as  may  best  suit  our  purpose.  And,  as 
the  figure  shows,  either  of  these  faces  at  will  may  be  made  more 
conspicuous  than  the  other  by  proper  selection  of  the  angles. 

We  have,  then,  a  system  of  true  ohlique  projection :  it  is  more 
flexible  than  the  isometric,  always  quite  as  easily  executed  and  in 
many  cases  more  so,  and  like  it  exhibits  the  three  dimensions  in 
one  view.  All  lines  lying  in  planes  parallel  to  the  paper  are  shown 
in  their  true  forms  and  relations ;   and  not  only  these  but  lines  per- 


FiG.  276 


pendicular  to  the  paper  are  shown  of  their  actual  dimensions,  the 
introduction  of  any  such  senseless  appliance  as  the  "  isometric 
scale  "  being  prevented  by  the  very  nature  of  the  process. 

346.  This  system  is  well  adapted  for  purposes  similar  to  those 
in  which  isometric  drawing  is  employed — such  as  the  representa- 
tion of  joiner- work,  as  exemplified  in  the  case  of  the  box.  Fig. 
276,  and  in  that  of  the  peculiarly  notched  and  fitted  pieces  shown 
in  Fig.  277.  In  the  illustration,  and  especially  in  the  sketching  of 
small  mechanical  details,  it  possesses  the  decided  advantage  over 
isometry  that,  as   shown   in   Fig.  278,   circles  whose   planes    are 


246 


DESCRIPTIVE    GEOMETRY. 


parallel  to  tlie  paper  are  represented  by  circles,  whicli  greatly  ex- 
pedites the  work  of  construction.  Those  lying  in  planes  perpen- 
dicular to  the  paper,  however,  must  here  too  be  represented  by 
ellipses :  since  each  circumscribing  square  is  projected  as  a  rhombus, 
the  axes  will  coincide  with  the  diagonals,  and  may  be  found  as  in 
Fig.  262. 


Fig.  277 


The  use  of  ordinates,  or  offsets,  in  determining  lines  whicli  are 
neither  parallel  nor  perpendicular  to  tlie  paper  is  substantially  the 
same  as  in  isometric  drawing.  Thus  in  Fig.  279  the  point /in  the 
plane  eg  is  located  by  setting  off  gli^  its  distance  from  the  plane  gn^ 
and  then  hf^  its  distance  from  the  front  plane  gm ;   the  2:)oint  h  is 


^^^zf 


Fig.  278 


determined  by  a5,  its  distance  from  the  end  plane  gn.^  Ixl  its  height 
above  the  plane  cg^  and  dh  its  distance  in  front  of  tlie  rear  plane, 
which  is  invisible.  The  two  points /"and  h  being  tlius  fixed,  the 
projection  of  the  line y^  is  determined;  and  the  rest  of  the  con- 
struction can  be  readily  traced  without  explanation. 

The  lines  which  cast  shadows,  and  are  therefore  to  be  made 


DESCRIPTIVE    GEOMETRY. 


247 


heavy,  can  usually  be  determined  by  inspection, — the  hght,  as  in 
the  common  orthographic  projections,  being  supposed  to  come 
from  over  the  left  slioulder,  and  to  go  downward  to  the  right  as  it 
recedes,  as  explained  in  (333). 

347.  There  is,  then,  no  need  to  pursue  this  subject  farther: 
the  principles  which  have  been  thus  briefly  set  forth  are  sufficient 
for  applying  either  of  these  modes  of  projection  to  any  subjects 
within  the  common  range  of  practice ;  and  additional  examples  of 
cavalier  projection  are  found  in  the  pictorial  illustrations  intro- 
duced in  the  preceding  chapters.  Both  are  very  useful,  with 
certain  limitations  which  have  been  suggested,  and  the  question 


Fig.  279 


whether  either  is  suitable  for  any  given  case  can  be  settled  by  ex- 
perience alone. 

But  one  thing  has  been  decided  by  experience  beyond  all 
question ;  and  that  is,  that  the  attempt  to  apply  either  isometric  or 
cavalier  drawing  in  the  construction  of  a  general  plan  of  any 
complex  machine  is  certain  to  result  in  a  melancholy  failure :  the 
distortion,  less  noticeable  in  the  case  of  minor  details  and  detached 
pieces,  becomes  unendurable  when  the  various  parts  are  assembled. 


PSEUDO-PERSPECTIVE. 


348.  For  the  purpose  of  producing  a  certain  effect  of  relief, 
and  conveying  at  least  some  idea  of  the  three  dimensions,  while  at 
the  same  time  avoiding  this  distortion  as  well  as  the  labor  of  con- 


248 


DESCRIPTIVE   GEOMETRY. 


structing  a  true  perspective  drawing,  a  mode  of  re23resentation  lias 
been  devised,  to  which  the  name  of  Pseudo-Perspective  seems 
appropriate,  of  which  we  give  a  single  illustration  in  Fig.  280. 

This  is,  in  principle,  a  modification  of  cavalier  projection ;  in 
that,  as  has  been  stated,  the  j^arallel  projecting  lines  are  inclined  to 
the  picture  plane  at  an  angle  of  45°.  But,  referring  to  Fig.  274, 
it  will  be  seen  that  if  the  cone  of  visual  rays  should  have  a  less 
angle  at  the  vertex,  the  projection  of  ca  would  be  shorter ;  and  by 
properly  choosing  this  angle,  the  projection  may  be  made  shorter 
than  the  line  in  any  desired  ratio,  while  its  direction  is  still  entirely 
arbitrary. 


Fig.  280 


The  pseudo-perspective  drawing,  then,  is  made  by  representing 
the  lines  which  are  parallel  to  the  paper  in  their  true  size,  while 
those  which  are  perpendicular  to  it  are  reduced  to,  say,  one  twelfth 
of  the  actual  length,  but  paratlel.  This  of  course  renders  tlie  result 
valueless  as  a  working  drawing ;  but  it  gives  a  sense  of  depth,  and 
such  representations  are  in  many  cases  well  suited  for  illustrations 
upon  a  small  scale,  such  as  cuts  for  encyclopaedias  and  the  like. 

The  distortion  in  the  true  cavalier  projection  is  due  to  the  men- 
tal recognition  of  the  facts  that  the  true  representations  of  receding 
lines  ought  to  converge,  and  that  equal  distances  upon  them  ought 
to  appear  less  as  they  recede.  Both  these  errors  are  made  less  con- 
spicuous by  reducing  the  lengths  of  these  representations,  which  is 
accomplished  by  the  method  of  drawing  above  explained. 


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